Starfleet registry numbers are all over the place in DSC and TOS and still somewhat inconsistent in the later series so I wouldn't put too much stock in a ship's launch year being in a particular number range.
USS VOY registry: NCC-74656
USS Protostar registry: NX-76884
There's a difference of 2 228 .
Not a huge difference when you consider the fact how huge UFP is and that its more than capable enough to build 1000 ships per year (and the fact the Dominion War started in 2373 and lasted until 2375 would have meant SF would have accelerated ship construction and research).
Caretaker episode stardate: 48315.6
If that's the case, we could infer the Protostar was initially built in late 2373 or very early 2374 and chosen as a testbed for new technologies (which it was testing while VOY was in the DQ - aka during the Dominion War).
Alternatively, the Protostar could have also been built and launched right after the Dominion War.
Its possible most SF shipyards were occupied making replacement ships lost in the War... but, SF also built the USS Prometheus during the War... so its not outside the realm of possibility the Protostar could be built during that time.
SF gets VOY logs, etc. in 2377, and is able to tinker with them which would allow them to finish and 'perfect' the Proto-Core at an accelerated time frame (Which was probably already in development). VOY returns in 2378.
7 of 9 could have joined and assisted in finishing the Proto Core at this point (she would have been debriefed with most of VOY crew, so she may have been asked to render some assistance).
Janeway is promoted to admiral by 2379 (Chakotay would likely be promoted to Captain at the same time or shortly after).
2380 - USS Protostar is refitted with the complete Proto-Core/Warp and Janeway Training hologram.
2381-2382 - USS Protostar is re-launched. Shortly in its journey, it encounters an anomaly and is thrown over 20 years into the past (around 2360-2363).
2383 - the kids discover the Protostar and flee Tars Lamora.
2366 (ish) - Gwyn is created (at this point the USS Protostar has been searched for years already).
Unless of course I'm completely and utterly wrong.

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