The question is why would Earth have such a large naval force?
True it is 60 years after WW III and that might be enough time...to rebuild after the Post Atomic Horror.
Nations have military forces because they require them, they serve a purpose. The reason for the forces would come into existence prior to the creation of the naval ships. If you have colonies, freighters, excursive trade routes that are endangered somehow, THEN
you build your navy. If you have a single colony and you round trip a civilian ship twice a year, then maybe you only require one naval ship. But as you acquire multiple colonies, dozen/hundreds of civilian freighters and lots of trading partners the pressing need for fleet develops.
So why does Earth need a naval force in your prequel?
The post-atomic horror wouldn't necessarily have been evenly spread across the Earth, entire nations and regions might have "sat out" the third world war. We know that at least one major missile base in Montana was untouched, and they tend to be primary targets. The Transamerica building in San Fransisco is still standing in the 24th century, perhaps indicating that the US was one of the nations that largely sat out, at least as much as possible.
That's a massive effort though that I don't believe is justifiably possible so soon after a WW. Not only does there have to be recovery but then a further organized effort to continue to the stars...colonize and then build individual national fleets for space......all after a world war of atomic levels...just 60 years after...I don't think so...that's too optimistic.
I don't think they sat out.
I think America Nuked China and the Arab Nations into Oblivion at the cost of few American Cities.
Some one needs to put the pieces of Trek's history together because it is reallly muff-ed up. That's what ENT should have done but that was a major mother-muffing-muff-up itself.