While watching Star Trek: First Contact, I noticed that Captain Picard keeps a case with models of all the previous Enterprises, including what seemed to me to be the Enterprise NX-01. This ship first appeared in the TV show Star Trek: Enterprise, which began in the early 2000s. When was the design of the ship first finalized to look the way it did? Had the design appeared in a previous episode that I was not aware of, or was this a preview to what we would se in Enterprise?
None of the ships in the case resemble the NX-01 except in as much as the NX was designed to resemble the ships already depicted.
The ships displayed are the Constitution class (TOS), A, B, C, D and E
Eaves himself states as to the NX-01's apparent absence:
"it hadn’t even been invented yet!!! this was at least 5 years before Enterprise came out. Sorry"
What of Star Trek Nemesis? One may be forgiven for thinking that since Star Trek: Enterprise had started production, perhaps the NX-01 was depicted as a gold model on the Enterprise-E in that film. Also - no:
Interview with John Eaves discussing Star Trek: Nemesis ships:
Moving ahead another couple of years, to Star Trek Nemesis, we are back to the observation lounge in what would be the Big E's final mission. The lounge set got a big face lift and the addition of a much-larger display area for the Enterprise models. Instead of six ships, this time there were going to be 12. For the second case, we built the Voyager, the Excelsior, the Grissom, the Reliant and under construction were the Phoenix and the Defiant. The deadline for these models moved when the filming scheduled changed the set order, and the observation lounge got moved up considerably. To make a long story short, the additional models didn't make it to the set and instead the E models were duplicated in pairs on either side of the viewing screen.
The Enterprise NX-01 was not affiliated with the United Federation of Planets like the other Enterprises, hence the prefix "U.S.S." (United StarShip or United SpaceShip), the NX-01 did not have a prefix, but in my opinion the NX-01 should have been U.E.S. (United Earth Ship); note: that the Fortunate Son and the Horizon had the prefix "E.C.S." (Earth Cargo Ship). That being said, when the Federation was formed, they started over at NX-01 and launched the U.S.S. Dauntless NX-01, with the "U.S.S." prefix. This should explain why when ever anyone mentions the original Enterprise, why they are referring to Kirk's Enterprise is because they are actually referring to the original Enterprise built by the Federation. Even the NX-01 was not the original Enterprise, there are plenty of Enterprises that came before that, such as the H.M.S. Enterprize or the shuttlecraft Enterprise OV-101 and of course the U.S.S. Enterprise CVN-65 which was once visited by Chekhov and Uhura in Star Trek IV: The Voyage Home.