This trip sounds absolutely great! My two favourite cities on Earth, bar none

Anyway, you have a week in each city so I would make the most of it. Having lived in London for (almost) all my life, I reckon it is important to stress that a week is not enough to fully appreciate these great cities and their zoos, but enough to see a good deal of it, and I hope you thoroughly enjoy both cities.
I have a few questions. Are you taking the Eurostar to get between the cities? It is faster in the long run (all the waiting at the airport + getting to and fro the airports) and a good deal more pleasant. Both Eurostar terminals in the respective cities are a delightful piece of architecture, with great transport links from them (especially in London), whereas the airports are fairly dull, crowded and isolated.
Also, do you plan to prioritise zoo-going or sight-seeing? I feel as if London and Paris both have so much to offer generally that it would be a shame to miss out on a day of it for Marwell, or Paris Aquarium. Neither nor are extraordinary collections, and I would echo previous suggestions to visit Hamerton, it really is an extraordinary collection and not too difficult to get to by public transport, although it isn't easy.
Parc des Felins is an excellent collection, with, notably for you from the states, Margays, Oncillas, Southern tigrinas are the interesting ones among an immense cat collection (with lots of subspecies of leopard other than Amur, as well

)
Marwell, while it is a good zoo, has nothing particularly special that makes it worth forgoing a day in Hamerton or exploring London for, while Hamerton is, in my opinion, well worth it but then again moderately difficult to access.
Howletts…. I don't know. Go if you really want, but what do you particularly want to see. It takes a full day from London, so another day gone. I also agree with
@Mo Hassan that London zoo is not worth it if you have 1 week. personally, I'd go to either Whipsnade or Hamerton (or both, although if I was faced with a similar choice, I'd go for Hamerton, Whipsnade doesn't have a very special species list). I'm sorry if I've just made things more complicated. Chester is also going to be rather difficult because it is really a full day zoo, perhaps even more if one arrives late. What are you planning in this regard? A very long day trip, or flying to London and then to Liverpool? Not sure how this would work but if it is feasible, Chester is well worth it and an excellent collection, as is Beauval, which seems to bring up a similar problem, especially in terms of public transport, by which it is only accessibly via a pretty cryptic bus service.
Paris zoo I would say is worth it. I would suggest going to the Paris Natural history museum (both the Grande Galerie and the gallery of compared anatomy and paleontology), the Menagerie and the Paris zoo in the same day. It is really a stretch but there is so much else to do in Paris... Alternatively you could try going to the Jardin des plantes and museums on a separate day to the zoo and seeing Notre Dame and surrounding attractions on the same day as the Zoo. Parc des Felins is a difficult one because you have a limited time, but if you really want to and like Disney, you could also fit in Disneyland on that day I guess...? I don't know how fast you are at visiting places, and what you would find interesting. I've never visited Disneyland Paris (not my cup of tea) but I think it might just be feasible if you cook up a very sturdy plan in terms of transport between the places. I'm wondering whether it might be worth renting a car for the last two days in Paris to see Beauval and PdF and wherever else.
Sorry if I'm rambling, I've just realised how long my post is! I get a little carried away with these things, these being by favourite two cities and all

. Don't hesitate to DM me if you want any help/advice as I know both cities quite well (a pair of grandparents live in one and I have lived forever in the other), so feel free to do so, I would be very happy to be of service, travel is my thing!
Hope your trip goes really well