For baptised Catholics who contract a civil marriage (neither having been married before), Canon 1160 applies. Their marriage is lawful but canonically invalid because of a defect of form as the proper canonical service was not followed.

1160. For a marriage which is invalid because of defect of form to become valid, it must be contracted anew in the canonical form, without prejudice to the provisions of Canon 1127 §2.

Canon 1127 §2 ostensibly applies to mixed marriages and allows certain things to be dispensed with, although "for validity, some public form of validation is required."

Thus a civil marriage between Catholics is validated by contracting it anew according to the rites of the Church. This is what happens in Germany as stated in the question; I've also attended a similar "civil+church" process in France.

For baptised Christians who become Catholics, retroactive validation is possible in the same way. For this to be available, consent must be present and persist and any impediment must have ceased. Such an impediment might be that there was a surviving former spouse at the time of the marriage: the Catholic marriage can be retroactively validated only as far back as the death of that former spouse. If consent was not present at the time of the marriage, retroactive validation only applies from when consent was subsequently given. This is covered by Canons 1161 to 1165.

This second process appears also to be what is followed if a Catholic divorces and then contracts a civil marriage. In this case, the second marriage is canonically invalid: it can't be conducted in church and is canonically bigamous. Such a marriage can be validated once the former spouse has died (and validated as from that point, not before it).

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My book of Canon Law calls the second process "Retroactive Validation". The Vatican translation online is more literal and calls it Radical sanation, "making it healthy from the root".

Natural Marriage

In the Catechism of the Council of Trent

The Catechism of the Council of Trent describes the difference between natural and sacramental marriage, which it describes as "two points of view," in its section on the sacrament of Matrimony:

Definition Of Matrimony. Matrimony, according to the general opinion of theologians, is defined: The conjugal union of man and woman, contracted between two qualified persons, which obliges them to live together throughout life.

…

Twofold Consideration of Marriage. … matrimony is to be considered from two points of view, either as a natural union, since it was not invented by man but instituted by nature; or as a Sacrament, the efficacy of which transcends the order of nature.

Marriage As A Natural Contract. As grace perfects nature, and as that was not first which is spiritual, but that which is natural; afterwards that which is spiritual, the order of our matter requires that we first treat of Matrimony as a natural contract, imposing natural duties, and next consider what pertains to it as a Sacrament.

…

Marriage Considered as a Sacrament. … Matrimony is far superior in its sacramental aspect and aims at an incomparably higher end. For as marriage, as a natural union, was instituted from the beginning to propagate the human race; so was the sacramental dignity subsequently conferred upon it in order that a people might be begotten and brought up for the service and worship of the true God and of Christ our Saviour.

In Pope Leo XIII's encyclical Arcanum

A sacrament is a conduit of grace.

As Pope Leo XIII wrote in his encyclical on marriage, Arcanum:

Marriage…is a sacrament, because it is a holy sign which gives grace

and

Marriage has God for its Author, and was from the very beginning a kind of foreshadowing of the Incarnation of His Son; and therefore there abides in it a something holy and religious; not extraneous, but innate; not derived from men, but implanted by nature. Innocent III, therefore, and Honorius III, our predecessors, affirmed not falsely nor rashly that a sacrament of marriage existed ever amongst the faithful and unbelievers.

Thus, marriages among unbelievers ("natural marriages") are sacred, too.

Two non-baptized who marry still have Origin Sin, which prevents their being able to receive sanctifying grace. So, while all valid marriages are sacred, only in Christian marriage is husband and wife able to receive sanctifying grace.

To your question: "At what point does a 'natural marriage' become sacramental?", I'd answer: When the spouses are able to fully receive sanctifying and sacramental graces.

Baptism, which takes away Original Sin, enables one to receive sanctifying grace. Valid baptism outside the Catholic Church (e.g., in a Protestant sect) would still take away Original Sin, but one is at least a material heretic by professing adherence to a heretical or schismatic sect, which presents a grave obstacle to salvation.

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Cf. also this passage from Arcanum, in which Pope Leo XIII explains that the contract and sacrament of Christian marriage are inseparable:

23. Let no one, then, be deceived by the distinction which some civil jurists have so strongly insisted upon - the distinction, namely, by virtue of which they sever the matrimonial contract from the sacrament, with intent to hand over the contract to the power and will of the rulers of the State, while reserving questions concerning the sacrament of the Church. A distinction, or rather severance, of this kind cannot be approved; for certain it is that in Christian marriage the contract is inseparable from the sacrament, and that, for this reason, the contract cannot be true and legitimate without being a sacrament as well. For Christ our Lord added to marriage the dignity of a sacrament; but marriage is the contract itself, whenever that contract is lawfully concluded.

A valid natural marriage becomes sacramental as soon as both parties are baptised. There is no need to do anything else after being baptised, either as a Catholic or in a denomination that observes the Catholic form of baptism (water and the trinitarian formula). If the baptism is not valid the marriage remains natural. The parties are the ministers of the Sacrament and exchange of consent is the essential element.

1621 In the Latin Rite the celebration of marriage between two Catholic faithful normally takes place during Holy Mass, because of the connection of all the sacraments with the Paschal mystery of Christ. In the Eucharist the memorial of the New Covenant is realized, the New Covenant in which Christ has united himself for ever to the Church, his beloved bride for whom he gave himself up. It is therefore fitting that the spouses should seal their consent to give themselves to each other through the offering of their own lives by uniting it to the offering of Christ for his Church made present in the Eucharistic sacrifice, and by receiving the Eucharist so that, communicating in the same Body and the same Blood of Christ, they may form but "one body" in Christ.

1622 "In as much as it is a sacramental action of sanctification, the liturgical celebration of marriage . . . must be, per se, valid, worthy, and fruitful." It is therefore appropriate for the bride and groom to prepare themselves for the celebration of their marriage by receiving the sacrament of penance.

1623 According to Latin tradition, the spouses as ministers of Christ's grace mutually confer upon each other the sacrament of Matrimony by expressing their consent before the Church. In the tradition of the Eastern Churches, the priests (bishops or presbyters) are witnesses to the mutual consent given by the spouses, but for the validity of the sacrament their blessing is also necessary.

1624 The various liturgies abound in prayers of blessing and epiclesis asking God's grace and blessing on the new couple, especially the bride. In the epiclesis of this sacrament the spouses receive the Holy Spirit as the communion of love of Christ and the Church. The Holy Spirit is the seal of their covenant, the ever available source of their love and the strength to renew their fidelity.

III. MATRIMONIAL CONSENT

1625 The parties to a marriage covenant are a baptized man and woman, free to contract marriage, who freely express their consent; "to be free" means:

• not being under constraint;

• not impeded by any natural or ecclesiastical law.

1626 The Church holds the exchange of consent between the spouses to be the indispensable element that "makes the marriage." If consent is lacking there is no marriage.

1627 The consent consists in a "human act by which the partners mutually give themselves to each other": "I take you to be my wife" - "I take you to be my husband." This consent that binds the spouses to each other finds its fulfillment in the two "becoming one flesh."

CCC on Marriage

Can. 1055 §1. The matrimonial covenant, by which a man and a woman establish between themselves a partnership of the whole of life and which is ordered by its nature to the good of the spouses and the procreation and education of offspring, has been raised by Christ the Lord to the dignity of a sacrament between the baptized.

§2. For this reason, a valid matrimonial contract cannot exist between the baptized without it being by that fact a sacrament.

Jimmy Akin

While Catholics are required to follow the required form of marriage non Catholic's are not

A defect of form case can only come into play if one of the parties to the marriage is a Catholic (who has not left the Church by a formal act). In a case involving a defect of canonical form due to a wedding taking place “outside” the Catholic Church, the Tribunal must be able to establish certain facts including the following: that the Catholic party was bound by canonical form; that the Catholic party had not left the Church by a formal act prior to or at the time of the wedding; and that the marriage was never subsequently convalidated or otherwise rendered valid in the eyes of the Catholic Church.

Defect of form