When Enterprise makes it back to Earth under fire, the ships that came to her aid had registries of NV and NW.
That's not true. None of the Earth Starfleet vessels (other than the Enterprise and the Columbia) had names or registry numbers printed on them. This comes directly from Doug Drexler. In fact, none of the ships other than the NX class ever even had official class names, ship names, or registries, even in a script or on concept art (with the sole exception of the "NC-27" for the Sarajevo-type ship, which was on the concept art but not on the actual CGI model). Any ship renders or artwork with such monikers as "NV" or "NW" are solely fan-created things.
And to the OP: The "01" in "NX-01" has nothing to do with the TOS Enterprise or even the Space shuttle Enterprise. TPTB chose it simply because it sounded kewl to them (and because they used it previously in Voyager for another ship, the NX-01-A Dauntless, which they though sounded kewl too; the two ships have nothing in common except the registry). Only later did they retroactively come up with explanations for it, like the "NX class" and "NX Project."