Jeff,

Here's what I can provide to try to help

In approximately which year did UG switch from the traditional style cases of the 1940s / 50s
to the case with the Speedmaster-style bezel? [and I do apologize for the Speedmaster reference]

There's no real swith to me but rather a smooth case design evolution based on the Polerouter designed by Gerald Genta and launched in 1954

Based on Pietro Sala's book looking at Compax and Tri-Compax sections,
we can see that "type of cases" with polished steel integrated bezel from 1960, although there might be some differences but the typical lugs will remain unchanged

here's a Compax from that era (maybe 1963/64)

they will evolve from that year to possibly 1965 the year it is believed the Compax was launched and used that specific case (later case is maybe thinner, I can't make measurements at the moment sorry, watches are at bank safe...)
The same case will be shared both for the Compax and late Tri-Compax up to the production of both lines of product will cease.

The bezel is not specific to UG,
it's been used on many chronographs from the Speedies to this unusual Ulysse Nardin V72 powered chrono just to name a few among many others

Did the transition to the Valjoux 72 movement occur at around this same time or did the V72 come later?

I believe the Valjoux 72 was first used for the Compax in 1965, it is said UG has stopped production of in-house calibers in 1968.
I don't think V72 have been massively used by UG as I only know those late Compax and Space Compax as UG V72 powered watches, maybe they're others ?

An interesting alternative movement used by UG was the Valjoux 23 which powered this late Uni-Compax (movement is signed 125)

no other V23 powered UG powered watches are known to me

When did we first see the true 24-hour indication chronographs, the Aero-Compax?

I'm guessing you're talking about this

this watch is a total mystery to me,
I've seen too many variations of dials/hands/bezel combos without ever seeing any reliable document to be able to make any attempt to help

All the above information have to be taken cautiously,
they are the result of what I have read here and there but I can't guarantee they're all absolutely right.

Maybe Lou can provide more.

Laurent