When does the bread and wine become Body and Blood?

Although I am not a Catholic I believe I can answer your question more determinately.

The only place where the exact instant seems to have been conjectured within Roman Catholic tradition is by St. Thomas in the Summa Theologica, Part 3, Question 75, ‘Article 7. Whether this change is wrought instantaneously?

And therefore it must be said that this change, as stated above, is wrought by Christ's words which are spoken by the priest, so that the last instant of pronouncing the words is the first instant in which Christ's body is in the sacrament; and that the substance of the bread is there during the whole preceding time. (St. Thomas in the Summa Theologica, Part 3, Question 75, ‘Article 7. Whether this change is wrought instantaneously?)

I am assuming that when he says ‘the last instant of pronouncing the words’ he means two instantaneous events. The bread turns into his body after saying ‘this is my Body’ and another event when the wine turns into his blood, after saying ‘this is the chalice of my Blood’. Reading the official documents are somewhat complicated because Greek philosophical terms about substances, their relative quantities and the accidents or things we experience from our sense when observing those substances are couched in the terms used to explain the intended doctrine. Basically in laymen terms, our senses perceive ‘accidents’ but only the mind knows the ‘substance’. What this essentially implies is that Catholics while still seeing the accidents of regular bread and wine that the substance of Christ is miraculously maintaining, this transformed substance. For example the transformed material is to literally understood as the physical body of Christ:

Christ and all the constituents of a true body, such as bones and sinews, but also Christ whole and entire. (Catechism of the Council of Trent)

However, since the notion of the Presence in the Mass is somewhat higher than each element it implies not just the body and blood but the soul and divinity of Christ is present, i.e. a complete Christ in both events. In other words, this higher presence of soul and divinity occurs when becoming the bread and then is some sense becomes more completed when the wine is transformed. The notion of a complete unified Presence can be seen from the Creed of Pope Pius IV.

I profess, likewise, that in the Mass there is offered to God a true, proper, and propitiatory sacrifice for the living and the dead; and that in the most holy sacrament of the Eucharist there is truly, really, and substantially, the Body and Blood, together with the soul and divinity, of our Lord Jesus Christ; and that there is made a conversion of the whole substance of the bread into the Body, and of the whole substance of the wine into the Blood, which conversion the Catholic Church calls Transubstantiation. I also confess that under either kind alone Christ is received whole and entire, and a true sacrament. (Creed of Pope Pius IV)

So now the question occurs, if the bread and wine is only the physical literal body of Christ, hidden behind accidents and quantities that present to our eyes a mere substance of bread and wine, how then is the Presence of Christ actually also sitting behind the veil of what our senses see? This answer is tied up in the term ‘concomitance’. The soul and divinity of Christ is understood as being present through another term called ‘concomitance’. This in effect means the bread is all of Christ including his divine presence and so it the blood.

The full development of all these ideas can now be understood by the more difficult terms used by Sty. Thomas where ‘accidents’, ‘quantities’ and ‘concomitance’ bind the whole doctrine:

Nevertheless, since the substance of Christ's body is not really deprived of its dimensive quantity and its other accidents, hence it comes that by reason of real concomitance the whole dimensive quantity of Christ's body and all its other accidents are in this sacrament. (St. Thomas in the Summa Theologica, Part 3, Question 76. The way in which Christ is in this sacrament ,Article 4. Whether the whole dimensive quantity of Christ's body is in this sacrament?)

The reason St. Thomas seems to have desired to determine a precise moment is in handling objections that might indicate truth to the doctrine of consubstantiation, where according to a Lutheran view the substance of the body and blood of Christ are present alongside the substance of the bread and wine. To refute the idea of consubstantiation, transubstantiation must occur instantaneously, otherwise even for 1/1000 of a second part of the bread must still be bread. Therefore, the change must occur so quickly in time that there is no measureable instant of a gradual change. This change must be therefore, according to St. Thomas at the moment the last words of the consecration are spoken by the priest. Whether this moment is at the time the last syllable is uttered or the time that the last syllable is heard among those present would probably be exact for the purposes of St. Thomas. If I had to guess he means when the last syllable is leaving the vocal cords through the mouth of the priest because depending upon the distance of those present each might hear the final sound at slightly different times, voiding the completely instantaneous concept.

Although St. Thomas is quite exact in his identification of when the bread and wine become the blood and body of Christ, he leaves the precise moment of when they return to just bread and wine uncertain. He seems to indicate the whole transformation is outside of normal physics causing regular moments of succession to cease in the transformation. This makes the time where the presence is no longer there not a proper ‘moment’ following another ‘moment’ along the science of time. In other words, the starting moment is identified as it is preceded by another moment, but the end, not being preceded by a proper moment is not exactly determinate. Or in his words:

Of this time no instant is to be taken as proximately preceding the last one, because time is not made up of successive instants, as is proved in Phys. vi. And therefore a first instant can be assigned in which Christ's body is present; but a last instant cannot be assigned in which the substance of bread is there, but a last time can be assigned. (St. Thomas in the Summa Theologica, Part 3, Question 75, ‘Article 7. Whether this change is wrought instantaneously?)

Jumping off a bit from fredsbend's answer ...

¹³⁷⁷ The Eucharistic presence of Christ begins at the moment of the consecration and endures as long as the Eucharistic species subsist. (http://www.vatican.va/archive/ENG0015/__P41.HTM)

Consecration, in general, is a sort of declaration of an objects use for a purpose. In this case, it's the super-significant declaration of the bread and wine as the body and blood of Christ. This declaration, as confirmed by the Catholic Encyclopedia, occurs when the priest says This is my body and This is my blood.

When we speak of consecration without any special qualification, we ordinarily understand it as the act by which, in the celebration of Holy Mass, the bread and wine are changed into the body and blood of Christ. It is called transubstantiation, for in the Sacrament of the Eucharist the substance of bread and wine do not remain, but the entire substance of bread is changed into the body of Christ, and the entire substance of wine is changed into His blood, the species or outward semblance of bread and wine alone remaining. This change is produced in virtue of the words: This is my body and This is my blood, or This is the chalice of my blood, pronounced by the priest assuming the person of Christ and using the same ceremonies that Christ used at the Last Supper. That this is the essential form has been the constant belief and teaching of both the Eastern and Western Churches (Renaudot, "Liturgiarum Orientalium Collection", I, i). (Catholic Encyclopedia - Consecration)

I don't know of any resources that pinpoint an exact moment at which transubstantiation occurs. But, given the nature of consecration in general, it's logical to assume that the moment of change occurs at the end of each meaningful declaration. Portions of the sentence (this, this is, and this is my) aren't complete declarations. Only, This is my body is a complete declaration.

So, taking the longer form for the blood, I would posit that the reality (not the substance) of the liquid inside changes slightly with each successive declaration.

Take this, all of you, and drink from it;

At this point, even before any change, "all" are commanded to drink from the cup.

this is the cup of my blood,

The liquid inside the cup is now and instantaneously Jesus Christ's real blood, fully and wholly.

the blood of the new and everlasting covenant.

The significance the blood in cup is now that of the Eucharistic covenant. Before, it was "just" Jesus' blood.

It will be shed for you and for all so that sins may be forgiven

Not 100% sure of a reality change here, other than in the mind. But, perhaps the blood in the cup is now transcendently linked to every other Eucharistic celebration and consumed for the forgiveness of sins.

Take this with a grain of salt though. I'm not a canon lawyer. But, I think this interpretation is consistent with the Church's teaching that the change is produced by the virtue of the priest's words acting en persona Christi.

Transubstantiation occurs during the consecratory thanksgiving during the single act of worship called the liturgy of the Eucharist.

1377 The Eucharistic presence of Christ begins at the moment of the consecration and endures as long as the Eucharistic species subsist.
http://www.vatican.va/archive/ENG0015/__P41.HTM

During Mass the liturgy of the Eucharist begins a particular time and follows this structure:

1346 The liturgy of the Eucharist unfolds according to a fundamental structure which has been preserved throughout the centuries down to our own day. It displays two great parts that form a fundamental unity:

• the gathering, the liturgy of the Word, with readings, homily and general intercessions;
• the liturgy of the Eucharist, with the presentation of the bread and wine, the consecratory thanksgiving, and communion.
  The liturgy of the Word and liturgy of the Eucharist together form "one single act of worship";170 The Eucharistic table set for us is the table both of the Word of God and of the Body of the Lord.
  http://www.vatican.va/archive/ENG0015/__P40.HTM

The consecratory thanksgiving is the moment when the priest holds up the bread and repeats the words of Christ: "This is my body ..." and so on. Likewise, he says the same for the wine "This is the cup ... " and so on.

1412 The essential signs of the Eucharistic sacrament are wheat bread and grape wine, on which the blessing of the Holy Spirit is invoked and the priest pronounces the words of consecration spoken by Jesus during the Last Supper: "This is my body which will be given up for you.... This is the cup of my blood...."
1413 By the consecration the transubstantiation of the bread and wine into the Body and Blood of Christ is brought about.
http://www.vatican.va/archive/ENG0015/__P44.HTM

Now one might ask: what if the priest does not finish the consecration for some reason? Say he faints. The wording of the Catechism does not address this directly except it does simply say this in the canon law:

Can. 927 It is absolutely wrong, even in urgent and extreme necessity, to consecrate one element without the other, or even to consecrate both outside the eucharistic celebration.
http://www.intratext.com/IXT/ENG0017/__P39.HTM

Seemingly, then, the consecration must be finished despite any circumstances. This bit of canon law also implies that it is possible to consecrate the bread but not the wine, however serious a sin it might be.

This further leads to another question: suppose communion, the last part of the liturgy, is not achieved -- is the bread and wine still Body and Blood? It seems yes because of what is stated in 1376. The "presence of Christ ... endures as long as the Eucharistic species subsist." Once transubstantiated the Body and Blood are Body and Blood so long as they exist. I assume then their destruction by eating or rot or other leaves them as not Body and Blood. However, not finishing the eucharistic celebration may be 'absolutely wrong' as well in light of the Canon Law quote above (I really cannot speak authoritatively on that).

Why does it occur at that moment and not some other? 1376 (below) states that it is "Because Christ our Redeemer said that it was truly his body that he was offering under the species of bread" at the last supper. So this statement argues both why transubstantiation occurs and when, and, subsequently, why it occurs at that time.

The definition of transubstantiation:

1376 The Council of Trent summarizes the Catholic faith by declaring: "Because Christ our Redeemer said that it was truly his body that he was offering under the species of bread, it has always been the conviction of the Church of God, and this holy Council now declares again, that by the consecration of the bread and wine there takes place a change of the whole substance of the bread into the substance of the body of Christ our Lord and of the whole substance of the wine into the substance of his blood. This change the holy Catholic Church has fittingly and properly called transubstantiation."
http://www.vatican.va/archive/ENG0015/__P41.HTM

I don't have easy access to older items, however, 1376 is a quote from the Council of Trent (1500 something).

SOURCES
Catechism of the Catholic Church
Code of Canon Law