Itinerary feedback and 2 for 1 Questions

I will try to help. It looks like you will get a lot of benefit from the 7-day Travelcard, as you will be using the transport feature at some point every day you are there. And you will get good use of the 2-4-1 offers on the 6 sites you plan to visit on Friday and Monday. Tuesday’s museums are free, except for the special exhibits.

You do have to wait in the ticket line if using the 2-4-1, so that is a slight negative. But we did not find that onerous in 2016. I know the Churchill War Rooms in particular have become much more popular since then, but perhaps the enthusiasm will abate somewhat by next year.

You can buy the Travelcard at Paddington train station (make sure you are at the train window, not the Tube). They may or may not have the voucher booklets available, so I suggest printing our the ones you know you want at home, and bringing them along.

You are correct that you cannot buy the Travelcard at Heathrow, so you will have to pay for your first journey into London. Heading to Paddington by Tube first sounds like a pain to me; you will have to make a transfer along the way and then backtrack to Earls’ Court after buying the cards. For GBP 12.10 each you can get 90-day in advance tickets on the Heathrow Espress and be at Paddington in 15 minutes. This is pounds more than the Tube but would be well worth it to me for convenience after a long flight. Once you have purchased the cards at Paddington you can use them to take the Tube to Earls’ Court.

You will need regular train tickets to Bath, and can save quite a bit on those by buying Advance tickets 11-12 weeks in advance. Start at this website, and it will take you to the correct one to purchase tickets.

http://www.nationalrail.co.uk

It is acting quirky today; I have seen complaints about that online and just experienced the problem myself. So wait a week or so to let it get sorted out.

The British Museum and National Gallery are both very large, and visiting them both in one day would be exhausting.

It's a well organized itinerary, though I agree with Philip about Thursday. The British Library doesn't take too long, but the other two museums could each take up a day. Decide on your top choice and give it the time you think it deserves, then go to the other if you still have the time and energy. (British Library would be my third choice. )

I agree with you about the Windsor + Bath tour, probably inadequate time in both. Again, pick one destination or the other.

Saturday may give you some free time, especially if you either skip the crown jewels or get to them as soon as the Tower opens to avoid the line. The exhibit on the bridge might take an hour if you really cover it. St. Paul's is a fairly short walk and might take two hours -- more if you climb the dome but you'll get a similar view from higher up at the Sky Garden.

You can walk past the fenced and heavily guarded entrance to Downing St. but you won't see much of #10.

Some thoughts based on our trip to London last September:
Friday: You might consider going to the British Museum. It is free so if you decide after an hour that you are just too tired and want to go back to your room, you have not spent any money on an entry fee and you're not committed to a tour. The museum is open until 8:30pm on Fridays. We only spent about an hour at the museum which was enough to see the Rosetta Stone and other wonderful exhibits on the first floor. FYI: To get to our hotel from Heathrow, we took the Heathrow Express to Paddington Station (about 20 pounds each) and then took the subway to the station close to our hotel. Very easy!
Saturday: We did this same itinerary. We took the bus to the Tower Bridge and then walked from the Tower of London to St. Paul's Cathedral. The Tower of London can get crowded so you might consider going there first and then to the Tower Bridge. We got to the Tower of London when it opened and went straight to the Crown Jewels. Then we went back to the moat area where we took the Beefeater tour (free). I don't recommend the Beefeater tour. I think it would be more beneficial to get an audio guide and take your own tour. There is a lot to see. We stopped by the Roman Wall that is across the street from the Tower of London and then on to St. Paul's. We walked to the top of the tower at St. Paul's which we really enjoyed (253 steps) but you definitely have to be in pretty good shape and not claustrophobic!

Monday: We enjoyed taking tours at both Westminster Abbey and Parliament (Palace of Westminster). Purchase a ticket ahead of time for Westminster Abbey and then sign up for the Verger tour as soon as you arrive. You get to tour several places inside Westerminster Abbey with a verger tour that others don't see. Same for Parliament. Since it was September, we could purchase tickets on site (across the street from Big Ben tower) but it is cheaper if you purchase tickets online. If you take these tours, you would have to rush to get to Churchill War Rooms.
Tuesday: You might consider getting tickets for the Churchill War Rooms first thing Tuesday morning, then head to the museums for the rest of the day.
Travel Card: We did not get a 2 for 1 cards. That sounds like a good way to save money. We did, however, get an Oyster card which we used for buses and subways in London. It was so convenient. You download an app on your phone so you can see the balance on your Oyster card and add more as needed. I purchased the Oyster cards at Paddington Station but I would order them online ahead of time and you will save some money. There is a booth at Paddington Station where they sell the Oyster cards and they are very helpful.