Let's first of all be clear about canonical form. A Catholic is normally obliged to celebrate their marriage before a priest and two witnesses, according to the rites of the Church. There are exceptions and dispensations. If a Catholic wishes to marry in a non-Catholic Christian Church, a dispensation from canonical form can be granted. In such a case, the non-Catholic minister solemnises the wedding. A Catholic priest may be present and say a prayer or whatever, but the wedding cannot be a joint celebration with two ministers solemnising it. (There is also provision for exceptions: when there is no priest available, a lay person can be deputed to solemnise marriage, and under certain conditions, if a Catholic cannot have recourse to an appointed minister, the marriage can be contracted before witnesses alone. These cases would not normally occur in England.)
Now if a Catholic marries without observing the canonical form, without a dispensation, and without it being one of the (rare) cases of necessity, the marriage is invalid in the eyes of the Church, according to current canon law. (Remember that it is the defined teaching of the Council of Trent that the Church can impose diriment (invalidating) impediments to marriage.) The most common cases of invalid marriages happen when a Catholic has married in a Register Office and not solemnised the marriage according to the rites of the Church. People in such cases often ask the priest "Can I have a blessing for my marriage?" The answer is "No, but we can put the marriage right." (though see my caveat at the end of the post.)
The two ways of putting the marriage right are convalidation and sanation. Convalidation, canonically the ordinary route, is where the couple come to the Church, and take their vows anew in the form required by the Church. Beforehand, the priest has to prepare the usual paperwork for marriage and obtain permission from the Bishop. The service can be a quiet and discreet one - there simply need to be two witnesses present in addition to the priest. With convalidation, the couple must make a new act of consent to the marriage.
Often it happens that one spouse (in practice, usually the husband) says that they do not want to go through such a ceremony. I such a case, an application may be made for a sanation - sanatio in radice ("healing in the root"). What happens is that the various details and documents are obtained and an application is made to the Bishop for him to validate retrospectively the consent that was already given. Even though it was given without the proper canonical form, the Church can retrospectively validate that consent. This reminds us of the principle that it is consent that makes the marriage, not the priestly blessing.
So is the priest acting contrary to the civil law when he does a convalidation? Relying particularly on the guidance given by Neil Addison, I would argue that he is not. In the combox of the post Luther, Trent and getting out of state marriage, Neil highlighted a legal problem with my suggestion of having the wedding in Church first, and then going to the Register Office if desired. He advised that this would contravene the Marriage Act (1949) and then added:
If however the Church were to say that the couple can only have a Church marriage after they have been through a Civil Marriage ceremony in the Registry Office then that would be legal because the ceremony would be regarded in law as simply a private religious blessing.Now it is not my point here to argue concerning the rightness of such an arrangement - simply to observe that it seems to be a fairly clear reassurance that the celebration of a convalidation is not contrary to English law as is often suggested. (In the case of a sanation, it is certainly not against the provisions of the Marriage Act because there is no additional ceremony.)
In pastoral practice, the convalidation or sanation of marriage is increasingly important. I often find, when booking the baptism of a baby, that the parents have married in a Register Office, on a hotel, on a beach in America or whatever. I gently point out to the Catholic party that the marriage is not valid in the eyes of the Church and that it is important to have he marriage put right because they they can return to Holy Communion. It is very often a surprise to the Catholic spouse to discover that they are not actually allowed to receive Holy Communion. The convalidation can often be fixed on the occasion of the anniversary of the wedding - this is a help in persuading non-Catholic spouses to co-operate.
(A pastoral note for priests - which most experienced priests will already be well aware of - is that before giving any encouragement about convalidation, it is essential to enquire whether either spouse was married before. If that is the case, then of course the second marriage cannot be convalidated except in the case that the first marriage is investigated by a tribunal and a decree of nullity is granted.)