Train stations

The train to Bath leaves from Paddington Station, while the train to Seaford or Eastbourne leaves from Victoria Station. The website seat61.com is very helpful regarding train travel in the UK.

Yes, London is ringed by about a dozen terminal stations.

A good place for you to stay (looking at your other posts) would be in the Kings Cross/Euston area.

Seven Sisters can be accessed by bus from either Brighton or Eastbourne Stations.

So you could take the fully accessible 24/7 bus #205 from Euston or Kings Cross to Paddington for Bath.

Thameslink trains run from St Pancras to Brighton for the Seven Sisters. Those Thameslink trains are in addition to the better known ones from Victoria highlighted above. At Brighton the buses to the Sisters stop right outside the station.

Euston and Kings Cross/St Pancras (across the road from each other) are only about 1/2 mile from Euston. So that is very handy for Harry Potter at Watford (another of your intended trips).

As you've noticed, the London train stations, for where you want to go, are spread out.

My suggestions is to stay in an area you find interesting for London and familiarize yourself with the Underground (Tube). It can easily get you to any train station. And if you could manage the NYC subways, you'll be able to handle the London Underground.

My suggestions that would include easy walking to many sites is the Victoria Station area. It also gives you easy access to the Victoria Coach Station that offers bus services to many places you wish to travel. And from Victoria Tube Station you can get to any of the other train stations in no more than 20 minutes.

If you use a smartphone, consider downloading the Citymapper app. It will give you a choice of transportation options in London including up to the minutes times.

All of the railways to London were built by private companies and they each wanted their own terminus.

The tube has improved its access for those with mobility issues in recent years but retrofitting lifts (elevators) into old stations can be difficult so it can take several lift rides to get from the platforms to street level. And, Not all stations have lifts. The TfL website has information on accessibility.

The tfl go app has a journey planner where you can pick the modes that you require, bus, tube, rail etc. You can also ask for a step free route.

The large stations are for long-distance journeys and each one serves a specific part of the country, so staying near one will not necessarily help with mobility. As other people have said only a few London Underground stations in central London are wheelchair-accessible, others only have escalators. Buses are probably more accessible for you.

If you intend to travel to Bath and the Seven Sisters (I assume the beauty spot on the Sussex coast, not the north-east London suburb location of the same name), you will want Paddington station for Bath and Victoria for the Seven Sisters.

Don't choose hotels based on train stations, but on a good area to be in. There seem to be 9 major stations and ironically Kings Cross and St Pancras are right next to each other.

Coming back from an Oxford day trip the return train I wanted was cancelled so I took the next train to London which went to Marylebone station and then just figured out how to Tube home. It all works.

We are flying into Heathrow and plan to spend the first 4 days in London.

If you did care to rearrange your trip plan a bit, there's a comfortable coach that goes from Heathrow to Bath. You might consider putting that at the beginning of the trip, then taking the train to London after your days in Bath are finished. That cuts down on one train trip out from London you'd have to take.

If you choose to take the coach from Bath to London, it will bring you into Victoria Coach Station.
Philip wrote, above, "you will want......Victoria for the Seven Sisters."
At this point, I refer you back to the post by Frank II. His advice is always excellent.
If you arrive by coach into Victoria Coach Station, you could easily stay in a hotel near there and leave out of Victoria Train Station.
(Not that you must arrive by coach to stay there...you can get to this area easily by tube once you're in London.)

I would also choose based on the comfort level of the hotel....comfy beds, shower with easy access for anyone mobility impaired, perhaps a good breakfast included in the price.
Remember you can always take transportation to the train stations from your hotel. If the tube and London buses are not for you, then perhaps a taxi/black cab.