At some point during VGR it was mentioned Starfleet had sent ships out to meet Voyager halfway with supplies and equipment. I don't remember if they specified if they were crewed or not, but that's my theory on where this ship came from. Autopilot set to rendezvous with Voyager and help it get back home. Like the Dauntless, except not a trick.
The Admiral said they redirected two deep-space vessels to rendezvous with Voyager... and if all went well, they were expected to meet them in 5 to 6 years (makes you wonder if in the original timeline those ships actually met with Voyager - but there was 0 mention of them pass that episode) - it suggests these 2 ships were already in deep space for a while... and I doubt that SF would send NX class ships (with experimental technologies that were still being tested) so far out. If something went wrong, SF would want to recover those ships with relative ease.
The registry number of the Protostar suggests the ship was built some time after Voyager disappeared.
Voyager's registry is 74656... Protostar's is 7688, and if they're indicative of how many ships SF has....
It would suggest that SF built 2227 ships before building the Protostar.
With UFP being massive in the 24th century (over 150 alien species in it), it could easily leverage each and every member world star systems (which would realistically have their own versions of Utopia Planitias - maybe not as big, but modern and mostly up to par) it could have easily built this number of ships within a year of launching Voyager (but it also depends on how many ships UPF builds every year and how old the Protostar is).
Technologically and resource wise, UFP could easily build thousands of ships all over UFP in a few months (probably less - all those replicators, transporters and computerized automation would definitely shorten construction times).
In peacetime, its likely SF is not building as many ships as it would during a war (or just after war - ala 'recovery period'), and I wouldn't put 1000 ships per year during peace time out of the question... during wartime, it would probably be a lot more (possibly along the lines of 5 - 10 times more when you factor in the premise that one drydock per each star system = 150 ships... if you have 10 drydocks per star system doing the work, that increases to 1500 ships... if you have 20 drydocks per star system, then that would be 3000 ships... and if you have 50 drydocks per star system, it would mean 7500 ships).
My guess is that all UFP member species planets would have comparable number of construction yard facilities. At the very least, they would all be up to spec and at minimum have about 20 - 50 drydocks each.
Its possible SF was experimenting with advanced post-Warp technologies (as I mentioned them above) after Voyager returned and something could have gone wrong in the process which resulted in the ship ending up in the DQ.
