If two Catholics, both free to marry in the Catholic church, desire (for a very compelling and honorable reason) to marry before their required pre-cana procedures are complete and do so in a non-denominational ceremony, may they then be married in the Catholic Church when their preparation is completed?
By "non-denominational", I take it you mean a civil marriage conducted by the State.
Canon 1055 has
§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.
It is the marriage contract which is the sacrament, and that can be effected anywhere. Naturally it's better that it is supervised by the Church (and Canon 1108 requires a Church presence for canonical validity), but the Church recognises civil marriages. It is for this reason that dissolution of a civil marriage is an impediment to re-marriage.
Canon 1108 has
§1. Only those marriages are valid which are contracted before the local ordinary, pastor, or a priest or deacon delegated by either of them, who assist, and before two witnesses according to the rules expressed in the following canons and without prejudice to the exceptions mentioned in cann. 144, 1112, §1, 1116, and 1127, §§1-2.
A civil marriage without an assisting clergy presence [which is actually forbidden in the UK — a civil marriage can contain nothing religious whatsoever] is invalid. It can be validated.
Validation is covered by Chapter X of the Canons on Marriage:
A marriage which is null because of defect of form must be contracted anew in canonical form in order to become valid, without prejudice to the prescript of can. 1127, §2.
Canon 1163 §1. A marriage which is invalid because of an impediment or a defect of legitimate form can be sanated provided that the consent of each party perseveres.
Canon 1161 §1. The radical sanation of an invalid marriage is its convalidation without the renewal of consent, which is granted by competent authority and entails the dispensation from an impediment, if there is one, and from canonical form, if it was not observed, and the retroactivity of canonical effects.
§2. Convalidation occurs at the moment of the granting of the favor. Retroactivity, however, is understood to extend to the moment of the celebration of the marriage unless other provision is expressly made.
§3. A radical sanation is not to be granted unless it is probable that the parties wish to persevere in conjugal life.
"Radical sanation" means "cleansing to the root" and validates the marriage from the time it was originally contracted (that is, in the civil ceremony). However, it's not something to be undertaken lightly, as it needs to be referred at least to the bishop and usually to Rome:
Canon 1165 §1. The Apostolic See can grant a radical sanation.
§2. The diocesan bishop can grant a radical sanation in individual cases [...]
Thus a civil marriage can be re-celebrated in church which will validate that marriage from that point, or with permission that re-celebration can validate the entire marriage from when it was originally contracted.
Anecdote: When I and others were received into the Catholic Church, there were two couples whose marriages were deemed to be slightly suspect because of a defect of form or other impediment, and that was a barrier to their participation in the sacramental life of the Church. A service was held at which they (and another couple, whose marriage was entirely OK but who wanted to express solidarity) effectively renewed their vows according to the canonical form approved by the Church. This was done privately, in the presence of a few close friends, but presumably it could have happened in a far more lavish way.
As Peter says in his answer, there is always an answer to the requirements of Canon Law: the Canons are intended to be compassionately just, and justly compassionate.
Necessary caveat and disclaimer: Nothing in this answer should be treated as a definitive answer on Canon Law or any individual case, for which the diocesan Tribunal should be consulted (the route to which starts with the parish priest).
I recently read about a priest (Orthodox, however) who is refusing to perform civil marriages. That's fairly interesting, but it brings up a good conundrum about the purpose and meaning of civil marriage nowadays. It shouldn't be this way, but it is becoming just a piece of paper to go with your real sacramental marriage.
As the commenters rightly said, you should (and obviously will) talk to the priest about it. You should do that now, as the "required pre-cana" is potentially not as required as you make it out to be. Maybe you could promise the priest you'll attend something like World Wide Marriage Encounter.
Also, there are two books you should read, one you're probably familiar with, which is Romeo and Juliette the plot of which hinges on a very real thing in the Church known as elopement. A priest, for a good reason, can get permission from the Bishop to marry you without the normal ceremony and in a place of your choosing (well, a sacred place at least).
The other book is The Betrothed in which a young couple is prevented from getting married by a priest, and their obedience to him prevents them from eloping causing lots and lots of problems.
Lastly, I'm trying to start a new Q & A site for Catholic that might be better suited to tackle these kinds of questions. This site is more for doctrine and dogma, and while the sacrament of marriage is touched on here, it more like you're asking for advice from fellow Catholics. The truth of the matter is, different priests are going to handle your situation differently and one priest is probably going to handle your situation differently than he would handle another person's similar predicament.
In the unlikely event that canon law comes up in a conversation with someone, keep in mind that the Church's law is supposed to be "healing" not "punishing" (that's what the priests on Relevant Radio say a lot at least). So if it feels like you're being punished, make sure you let the one who you think is punishing you know that, because in truth, you shouldn't be punished for wanting to do the right thing.