IIIClovisIII
Perhaps I was a bit hasty in writing my OP. My point really is, assuming it's reasonably in line with mainstream beliefs would they really feel comfortable adding it on to the existing canon? Who would and who wouldn't.
If you're interested in developing the "what if?" then there are "close" cases you can look at, and see how churchmen have in fact behaved.
Thomas, whose only nearly complete example is a Coptic translation, is heavily larded up with bullshy Gnostic additions. There does appear, however, to have been a "core"
Thomas that could be as old as any of the canonical Gospels. If that were so, then a clean core
Thomas would be a good bet to be the manuscript discovered in your hypothetical come-to-life.
Likely-core
Thomas does present a specific doctrinal problem. Saying 113 attributes to Jesus a present-tense kingdom lesson. Parallel teaching is in
Luke 17:21-22, but there Jesus addresses the Phraisees and
immediately tells the disciples about the usual future kingdom. In
Thomas 113, the present-tense kingdom lesson is addressed to the disciples.
There are also interesting cases of different churches' handling of known material. Some African churches and the Syriac church have modestly different canons from the familiar 27. Another window into "canon thinking" is to follow the fate of confidently authentic, possibly as early as
John or
Revelation, and painfully orthodox material like
1 Clement or the more credible letters of Ignatius of Antioch.
The significance of canonicity varies by church. Obviously, for
sola scriptura Protestants being in or out of the canon is night-and-day. For apostolic succession churches (including the Anglican Communion, Roman Catholics and Eastern Orthodox, plus some smaller communions), the canon is typically already supplemented for doctrinal purposes by patristic writings and even by ancient liturgical material, so, while being in the canon does impart some status, it just isn't make-or-break.
It's hard to imagine any strong pressure to include any newly authenticated material into the canon. If you think about it,
sola scriptura churches are committed to the existing canon, so anything else either conforms with the canon, and so is redundant, or else it's wrong. For the majority that isn't
sola scriptura, if something's available for study, then it's available for study. There's already plenty of stuff to read in church, and they don't use all of that as it is.