Planning England and Paris Itinerary Advice Greatly Appreciated

I have done the National Gallery, in the same situation as you, and lasted about 1 1/2 hrs. Given the location of your hotel, I think this is the only thing you should try to see on the day you arrive. I have never done the HOHO bus(I have found the ones in Bath and Edinburgh to be quite useful), but people say that with all the traffic in London(and the wonderful subway system), it's more frustrating than fun.

Have you considered tea as your dinner for one night? I have been to the ones at Harrods, Wolseley, and Fortnum and Mason. FM was the best. We did it on the day we arrived(it's walking distance from your hotel) and enjoyed the peaceful ambience. However, other people say that would put them to sleep if they did it on the day they arrived.

Just a few thoughts. Your hotel might store luggage for you and also might a!low an earlier check in. Worth asking the hotel. Hampton Court is super easy to do on your own from London. No need for a paid tour. Have you booked airline tickets? No need to back track to London. Fly open jaw USA to London, train to Paris and home from Paris. Even if already booked perhaps airline would allow a change of course with a fee. Worth checking it out. Good luck.

We took a food tour in Paris and really enjoyed it. We took the https://parisbymouth.com tour but I'm sure there are others.

"Day 3: London Bridge Tour-Tower of London Tour-Tower Bridge-St Pauls Cathedral self walking tour-Dinner at Sky Garden
Is there anyone of these activities that you suggest not getting the ticket or tour for and just opting for a walk-by or taking the tours for all? What is a suggested lunch place in this area?"

I recommend pre-booking your tickets to the Tower of London and arriving right when they open. If you enter as soon as they open, head straight to the Crown Jewels and see those before the crowds arrive. You'll be able to see the Coronation Regalia up close. Beefeater tours (free/included with the price of your tickets) leave every 30 minutes from just inside the entrance. Those give a great job of giving an overview of the Tower and its history. Plus, they're led by actual Beefeaters and Tower residents, not just tour guides. You can't pre-book those tours, just show up.

When we were in London in December 2022, after our visit to the Tower, we walked across Tower Bridge and ate at The Ivy Tower Bridge and thought it was delicious and reasonably priced for the location.

We've done the dome climb and Rick's self guided walk through St. Paul's and never felt the need for an additional guided tour.

I don't know that a sightseeing pass for London will save you any money. The British Library and National Gallery are both free. Time you spend seeing them is time you will not be taking advantage of one of the sights covered by whichever pass you buy. I don't know whether all the other places you mention in London are covered; they may be.

I am also doubtful about the value of any sort of rail pass for what you plan to do in England. It's quite possible it will be cheaper simply to buy tickets. I hope one of your more knowledgeable posters can provide guidance on that score.

You really can't see much of the Cotswolds via public transportation, certainly not "some of the best villages" on a day trip from Lo, and we've been told the taxis tend to be booked up by locals (for things like medical appointments), so you really can't count on filling transportation gaps with taxis unless you try to lock down a rigid schedule in advance.

There are good one-day, small-group (van) tours of the Cotswolds offered from Bath and from Moreton-in-Marsh. It's possible to spend the night in Oxford and get to Moreton-in-Marsh by train early enough to catch a tour originating there. I was very happy with my GoCotswolds tour from Moreton-in-Marsh. The MadMax tours departing from Bath also get good reviews. A tour will allow you to see some of the beautiful countryside and provide brief visits to a good number of the cute towns.

By traveling independently to Hampton Court rather than taking a tour there, you'll save enough money to cover a fair chunk of the cost of a Cotswolds tour.

For me Piccadilly Circus is a place you walk through on the way from one point to another, not a place to spend an afternoon. As an alternative, I'd suggest checking the walking-tour schedule on the London Walks website (https://www.walks.com/. There are many tours offered each day, starting from a variety of Underground stations. The cost is very low (15 GBP for adults and 10 GBP for students) and the guides are licensed professionals. Those tours are one of Europe's greatest bargains.

There are no useful rail passes or railcards for the amount of train travel you have planned. You might be able to save some money with Advance tickets if you can fix your times.
Oyster cards are no longer needed in London. You can just use contactless cards or phone or smart watch with no need for Oyster or physical tickets. Some people seem to prefer buying Oyster cards to avoid having to get credit or debit cards out but that will cost you an extra £7pp and personally I don’t see the need.
HoHo services are a poor idea in London. The traffic means that it takes a long time to get round the route.and if you get off it might take a long time for the next bus to come along and often you can’t get on. On top of that doing it on your arrival day is a recipe for falling asleep with jet lag.
If you are in Brighton it seems daft to me to trail isn’t across to the other side of London for two nights in the Cotswolds. The area is pretty but no prettier than the countryside in SE England close to where you will be e.g. the South Downs, New Forest etc etc. Unless you have a personal reason to go to the Cotswolds I wouldn’t make life difficult for yourselves.

Moreton in Marsh will be 3 1/2 to 4 hours from Brighton.
About an hour into whichever London station, allow 1 hour for cross London connections, then about 1 1/2 hours from Paddington.
That should be able to be done on 2 advance fares, one into London, the other London to Moreton.
Moreton is better than Stow for public transport - see a recent post on that.
Hampton Court is a London zonal fare, and Windsor a straightforward day return fare.
The issue of oyster vs contactless has been discussed to death on this forum. For various well rehearsed reasons some people prefer Oyster.

Your hotel will probably store your luggage for you if you can't check in early. I've never had a problem doing that.

I agree with the poster who suggested flying open jaw, arriving in London and leaving from Paris. No need to backtrack.

Some people like the Hop on Hop Off Bus. We did it on the day we arrived, but were nodding off most of the time. We were just riding however. If you actually get on and off you might not have that problem. But I've heard that taking regular buses is much cheaper and faster.

Have fun!