Adding to what you guys have said, I imagine the modern multipurpose 'starship' probably rose out of the limitations of each of the members' fleets that initially made up the fleet and the limitations in juggling deployments based on capabilites. For example, the Tellarite ships are slow, the Vulcans are poorly armed, the Starfleet ships had no shields, the Andorians had no science labs, and so on. So Admiral Smartguy says 'We need a ship that doesn't have any limits!' and behold, the modern Class One Starship is eventually born.
I see the cooperation as limited at first - or rather, forced. I imagine the combined command structure is basically there from the joint war efforts and from the Coalition of Planets, it just required some post-war refinement. (I'm not sure what fleet numbers we could infer, but I'm guessing the combined UFP fleet would be around 500 at most.) We would then have joint fleets deployed to various locations based upon ship-ability, in turn defining mission roles such as cruiser, scout, and science ship. So when they begin commissioning ships at first, its new ships that fill the same roles originated by each race's ship's abilities and limitations, building towards having those AND the multi-mission cruisers.
I think shipbuilding probably slid back from the ambitious NX-era designs towards simpler, cheaper, more reliable ships such as
Daedalus and then builds back towards more ambitious designs. I see
Daedalus as being the first ship built of combined technology between all the races, and the template for modern ship design, if not nearly as capable as, say,
Constitution class ships, and by no means a full multi-mission starship.
One nagging question is this: why does it seem like Earth starship design dominates Starfleet? If primary hull/secondary hull/nacelles is a human thing before the UFP, why does it seem to continue unimproved... advanced, but unimproved. I infer that it was the superior configuration (for various reasons of economy, efficiency, etc.), and that Starfleet simply combined all the technologies into and refined that basic form. I guess it depends on where you think
Daedalus falls on the family tree too. You can justify it as a pre-UFP cooperation project, or it could in fact be an Earth ship. Maybe the Coalition of Planets solves that issue though... the
Daedalus was a combined tech Coalition experiment commissioned as part of the war effort, before UFP founding, but is immediately absorbed into the interstellar Starfleet?
Where then do the Andorian ships go, and the Vulcan ships? I'd say they are retired as they are replaced, but constituent worlds also maintain small non-Starfleet reserve armadas, and in turn there are all-alien Starfleet ships such as
U.S.S. Intrepid from TOS being all-Vulcan. There would also perhaps be an all-Andorian
U.S.S. Shran.
I'd say that there was probably a bigger, better version of the basic
Daedalus design commissioned in the 2180s that was maybe rated as a Warp Seven engine. (Let's call this the
Horizon class.) I imagine that by 2210 the saucer design replaces the sphere of
Daedalus with probably NX-1100 which I'll call
Essex class for no real reason. By this time I'd also say that shuttles have been replaced as primary transit by multi-person transporters. Then by 2245 you have the first
Constitution class ship. Phasers and photon torpedoes don't arrive until about 2260, completing the 100 years.
