Lower Decks made it clear the uniform they used was just for the California class.
Well no, because the Parliament Class and Douglas Station officers are wearing the same uniforms as the Cerritos crew.
Not at the start of the show. In the first season everyone was wearing them, including starbase personnel and the crew of the Parliament class Vancouver.
We do see the Titan's crew wearing the First Contact uniform in Season 1. There may have been an earlier appearance, but I honestly can't remember.
Compare that to the 2380s in NuTrek, when we also had three shows running more or less concurrently: Lower Decks, Prodigy, and the Picard flashbacks. Each show used its own uniforms and different insignia. There was no cohesion—everyone was brewing their own soup.
None of those shows overlapped in-universe however.
LDS ends a year or two before Prodigy season 1 starts, and Prodigy ended at the same time as the Mars attack, but everyone (except the Prodigy crew) were all wearing the Picard flashback uniform, so there was no real contradiction there.
It's not like the TNG era shows didn't have a unified uniform. DS9 crew had their own unique uniform, before Generations (which was originally going to have another new uniform) and Voyager used it.
And it was meant to be assignment based, because Sisko was wearing the TNG uniform at the start of the Emissary, and switches to the DS9 uniform before the end. Nearly every Federation ship crew that visits the station is still wearing the TNG uniform, and when Sisko is reassigned to Starfleet Command by Admiral Layton he switches back to the TNG uniform.
So I absolutely have no problem with there being multiple uniforms in use in Starfleet at once.
And it's not like the different modern productions don't talk. Jack Crusher (junior) is mentioned in Prodigy, and Beverly's jacket is the same one she wearing in Picard Season 3. She's also in London, which was mentioned in Picard Season 3.
One of the Prodigy writers mentioned that Daystrom Institute packing up the portal tech was meant to be a connection to the portal tech in Picard Season 3, even if it didn't look similar, but Prodigy S2 was in development before Picard Season 3, so they probably wouldn't have known what it looked like.
The three Toronto based Live Action shows also shared crew members. I know Timothy Peel worked on all 3 (and I think Picard Season 1, but I'm not sure) shows as art director for the motion graphics (animated computer screens and 2d elements like that). Plus the same makeup team was on all three, and costume designers.