In TUC, on various charts and displays, including the Operation: Retrieve plans and a mission assignment readout on the Enterprise bridge, there are several starships represented. Here is a basic list:

U.S.S. Ahwahnee NCC-2048
U.S.S. Challenger NCC-2032
U.S.S. Constellation NX-1974
U.S.S. Eagle NCC-956
U.S.S. Endeavour NCC-1895
U.S.S. Enterprise NCC-1701-A
U.S.S. Excelsior NCC-2000
U.S.S. Helin NCC-1692
U.S.S. John Muir NCC-1732
U.S.S. Kongo NCC-1710
U.S.S. Korolev NCC-2014
U.S.S. Oberth NCC-602
U.S.S. Scovil NCC-1598
U.S.S. Springfield NCC-1963
U.S.S. Whorfin NCC-1024

Now, here’s the info that we already know definitely:
1. The Constellation is the class ship of the Constellation-class.
2. The Eagle, the Endeavour, the Enterprise, and the Kongo are Constitution-class.
3. The Excelsior is the class ship of the Excelsior-class.
4. The Oberth is the class ship of the Oberth-class.

The following are completely unknown to us:
1. The Ahwahnee
2. The Helin
3. The John Muir
4. The Scovil

Now, take a look at the remaining ships:
1. Challenger
2. Korolev
3. Springfield
4. Whorfin

Notice anything? They’re all the names of ship classes seen or mentioned in The Next Generation. (With the exception of the Whorfin-class, which was seen in Generations.) Is this a coincidence? I personally don’t think so. I think that these ships from TUC are in fact the class ships of the Challenger, Korolev, Springfield, and Whorfin-classes, respectively.

My evidence:
1. The numerous other hintings and references in TUC to TNG. (Such as the character of Colonel Worf, the use of Khitomer as a location, Scotty’s comment about “buying a boat,” [a possible reference to the Jenolen] and numerous others.)

2. (Goes along with 1.) The featuring of class ships of other classes seen in TNG and elsewhere, including the Oberth and Constellation.

3. Just the fact that it’s a bit of a stretch to think that all four vessels just happened to have names of future classes.

In my opinion, it’s fairly clear that the producers of the film were very keen on loading it with connections to the later time period, (after all, the movie was supposed to be a passing-of-the-torch thing) and that these ships are probably intended to be the class ships for the ship classes of the same name.

As far as regs:

Miranda-class
18xx to 38xxx

My guess is that Starfleet simply decided to name future classes Challenger, Korolev, ect. ect. and that there really is no story behind this whole thing.

The Challenger and Springfield classes both use scaled-down Galaxy saucers as a starting point. The elliptical saucer didn't debut until well after the turn of the century. Furthermore, the Challenger uses Galaxy-style warp engine nacelles. These classes date no earlier than 2320, and probably much later than that.

The Korolev may or may not be the class ship of the Goddard. If this is as long-lived a class as the Oberth or the Miranda, they could still be making 'em in the 2330s. The Goddard was acting as a transport, so it's likely they're not as susceptible to obsolescence as front-line starships.