At what point does a convert become a Roman Catholic?

This answer is to complement @PeterTurner's answer (which highlights how the process started in the heart) and @KenGraham's answer (which highlights the beginnings of a public profession). My answer emphasizes the culmination of the pulic profession: "a Catholic in good standing" / "practicing Catholic" that has to be maintained by remaining in communion with the Bishop of Rome.

As a Protestant who is doing research on pros and cons of converting, from the point of view of the Catholic church, you become a practicing Catholic once you celebrate your first communion which in a real way makes you "in communion" with the bishop of where you are residing, with the Pope, and by extension, with all Catholics who similarly declare such communion, whether living, in purgatory, or in heaven (communion of the saints).

Usually the first communion is preceded with:

• having gone through RCIA which includes:

  • assessment of whether you need a conditional or a regular baptism (which in your case, coming from the Anglican church, you most likely do not need, if you can procure your certificate of baptism)

  • sacrament of reconciliation

• on Easter, along with your RCIA cohort at your local parish, going through the profession of faith and the sacrament of confirmation, which preceded your first communion

From Wikipedia article on Conversion to Christianity — Catholicism (emphasis mine):

The Catholic Church considers all forms of baptism with water, including full immersion, affusion, and aspersion, that are done in the name of the Trinity as valid.[29]

Protestants (Lutherans, Moravians, Anglicans, Presbyterians, Baptists, Assemblies of God, Methodists, etc.) who convert to Catholicism are usually not baptized, but instead are asked to make a simple profession of faith at Mass on an ordinary Sunday. Confirmation usually follows (though not always), and the convert proceeds to receive first communion.

Eastern Christians (Eastern Orthodox, Oriental Orthodox, and Assyrians), are only asked to make a simple profession of faith and then begin participation in the Eucharist without having to be confirmed due to the Catholic Church recognizing eastern Christian sacraments.[31] Eastern Christians who convert to the Catholic Church are automatically enrolled into the eastern rite corresponding to the Church they originated from regardless of what sui juris Church they entered the Catholic Church through.[32]

The amount of instruction before reception varies depending on how active the person has been in their Christian life, and how informed they are about the faith. Validly baptized persons coming from previous denominations do not have to be enrolled in RCIA because the Church does not consider them catechumens since their baptism has already made them Christians. Private instructions may be given by a priest, which can last from a few weeks to a few months at most. After instructions have ensued, the person may be asked to pick a sponsor for confirmation if the pastor decides to perform the sacrament.

There may be a point later on when one becomes a Catholic in Good Standing, but the point of conversion comes much earlier in on the process. It's a discernment process which, for adults, I believe is similar to what priests go through:

It is impossible to be just to the Catholic Church. The moment men cease to pull against it they feel a tug towards it. The moment they cease to shout it down they begin to listen to it with pleasure. The moment they try to be fair to it they begin to be fond of it. But when that affection has passed a certain point it begins to take on the tragic and menacing grandeur of a great love affair.

-G.K. Chesterton (“The Real Obstacles,” The Catholic Church and Conversion)

That's the moment which a person "becomes" a convert. The Catholic Church has always taught that everyone is saved through her and everyone can be saved through her,

171. What is the meaning of the affirmation "Outside the Church there is no salvation"?

This means that all salvation comes from Christ, the Head, through the Church which is his body. Hence they cannot be saved who, knowing the Church as founded by Christ and necessary for salvation, would refuse to enter her or remain in her. At the same time, thanks to Christ and to his Church, those who through no fault of their own do not know the Gospel of Christ and his Church but sincerely seek God and, moved by grace, try to do his will as it is known through the dictates of conscience can attain eternal salvation.

CCCC 171

therefore everyone is more or less a member of the Church to begin with. So membership isn't as important as the process that begins in the heart, which is conversion and makes its way into a formal process through Baptism (for the unbaptized convert), and the other sacraments of initiation (Eucharist and Confirmation).

At what point does a convert become a Roman Catholic?

A non-Catholic Christian convert who has been validly baptized becomes a Catholic at the moment he makes a public profession of faith in the Catholic Church. This act of faith then allows you to receive the other Catholic sacraments including confirmation and the Eucharist.

The rite of reception of baptized Christians into the Communion of the Catholic Church states that "one who was born and baptized outside the visible communion of the Catholic Church is not required to make an abjuration of heresy [publicly] but simply a profession of faith.(1) Today, normally, an abjuration of heresy is made in the privacy of the confessional, though in the past it was often a public matter. After joining with the congregation in reciting the Nicene Creed, the person being received into the Catholic Church makes the following profession of faith:

I believe and profess all that the holy Catholic Church believes, teaches, and proclaims to be revealed by God.

As indicated in the Rite of Christian Initiation of Adults, adults joining the Catholic Church were formerly asked to abjure the previous faith to which they belonged ("Hebrew superstition", the Islamic "sect of the infidel", or "the heretical errors of the evil sect" from which they came). The profession of faith used was the Tridentine Profession of Faith. - Profession of faith

A Profession of Faith publicly made, usually before a Catholic priest, is sufficient to make you a Catholic, regardless if you receive the sacrament of confirmation and/or the Eucharist.

Prisoners on death row have made Professions of Faith into the Catholic Church, yet they were unable to receive the sacraments of confirmation or the Eucharist. They are definitely considered Catholic converts. I have stood beside a priest, who has received such a Profession of Faith. Immediately afterwards they make their first confession!

A first confession should be coordinated in normal circumstances between the priest and the catechumen.

There are also special considerations for receiving Orthodox Christians into full communion with the Roman Catholic Church.

Normally, a baptized Orthodox Christian is to be received into the corresponding Eastern Rite of the Roman Catholic Church (Decree on the Catholic Eastern Churches 4). However, if the newly received person is married to a Catholic in a different Rite (e.g. Roman), he or she may either remain in the Eastern Rite or transfer to the Rite of the spouse. (canon 112.2)