During my UK trip in April 2024, I visited London, Battersea, Whipsnade, Hamerton, Old Hurst, all from London within about a week, and then did Chester, Birmingham and Yorkshire all while in Sheffield for an event. Colchester, Cotswold, Hanwell, and Monkey World were considered but did not work out. I've been hoping to do a trip thread at some point.
Old Hurst was mostly there as a bonus before visiting Hamerton the same day and out of interest in a few species, so probably skippable unless you're particularly interested in crocodilians, raccoon dog or sloth bear. Battersea is small, accredited, and very child-oriented, but it is very accessible from London and has Scottish wildcat as the main unique species. It's far from a must-see otherwise and probably the last zoo of those listed I'd make a return trip, so I would mark it skippable, but if you're going to be close by, it's a way to kill a few hours, add a facility, and I liked the Scottish wild cat enough.
Hamerton is BIAZA accredited and has a lot of unique species. It is a bit difficult to get to but I would say fairly worth it in my opinion. My experience was really positive there.
London and Whipsnade are both doable on public transportation and well worth it. London has a bad reputation here due to the sense of decline from locals, but has some solid enclosures, history around every corner, and has a number of species not common in the US, particularly in the Land of the Lions complex. It's also easily doable in a day. Whipsnade is doable in a day if you have a car with you and has some massive, gorgeous enclosures and is especially attractive if you like hoofstock, some of which are not common in the US. Contains a great collection of larger species as well and some wonderful habitats.
To be quite honest, given that I'm going to be juggling at the very least graduate school work and an independent research project, I doubt I'll really be able to do anything longer than like a day trip.
If you are only doing a day trip, I do not recommend Chester for that criterion. It's an incredible facility well worth a visit, but I commuted from closer Sheffield and it was still a bit rough (there were delays in both directions and Easter holiday packing) and it really is a day-plus zoo again with the opening of Heart of Africa. I speedran almost everything else and I sort of regret only doing the one day It's better than nothing for sure, and that's not to say it isn't amazing but I really recommend holding off until you can do a fuller trip.
I can't recommend Birmingham or Yorkshire for your purposes either. The former is just too long a trip from London for too small a zoo, no matter how much it punches above it's weight-class in overflowing with smaller species. Yorkshire is more 'worth' a day trip in theory, and similar in character to Whipsnade with large enclosures and a megafauna focus, but a three hour commute would be pretty rough for a day trip.
In summary...
Visit - London, Whipsnade, Hamerton
Skippable - Battersea, Hurst
Not Fit for a Day Trip -- Chester, Yorkshire, Birmingham