Spells? How do I address the fact that water and fire, lightning and cold just don't mix? Yeah, I could handwaive it as "it's magical" but I really don't want to. Are there good alternatives?
The same general tricks that worked back in the day should work now. Fire just doesn't work underwater, cold has a reduced range/area and might create a nice big chunk of ice in the area, lightning changes from whatever it shape it was supposed to be into a burst.
Playable races? I'm not a particular fan of either monstrous races or "underwater humans with blue skin"
There's water genasi, I suppose. You could have a race of Atlantean/Lemurian style humans who don't breathe water or have blue skin, but do have skills & equipment adapted to living underwater, either magical or fantasy-tech type solutions - like the scuba gear made from giant shells in the old Mysterious Island, for instance.
and I'm particularly lost as to what would be good underwater variants of gnomes and halflings. Sure, I could leave them out, but I'd like to have something.
There are sea elves, so I suppose you could have sea-gnomes, who are also fey. If you use the 3e-gypsy-style halflings, they could be wandering in ships instead of caravans; if you use the traditional hobbitsy-borrowing halflings they could live in comfortably-renovated air-filled gaint-monster-shells, like communal hermit crabs.
Radiant: I mean, it's light, right? So will water diffuse it in some way?
Murky water perhaps enough so to make a difference, sure.
Cold: Will a cold spell freeze water? Could a ray of frost even travel underwater?
Yes & yes. Shorter ranges & smaller areas, but leaving ice behind. Could be interesting/different.
Poison: Would Poison Spray be diluted underwater?
Yes, then again, it becomes an 'inhaled' toxin, since it'd be absorbed by the water-breathers in the area.
Acid would presumably be 'washed off,' too, though, it might be deadly by damaging gills.
Thunder: Water is a better conductor of sound than air, so would that affect spells like Thunderwave?
Seems reasonable. It might create an actual wave.
I once had a DM who kept getting caught up on the science of how things should work, and pretty soon we were having to deal with convection and the "crushing depths of the sea floor", among other things.
It's a valid style, and one D&D has certainly tended towards in some ways at some points.
But, so is ignoring science and psuedo-science implications and just letting fantasy be fantasy. Factoids like "you can't get enough air breathing through a straw to remain conscious" or details like the specific heat of water or whatever can be safely ignored.