4 Days in London - does this itinerary look doable?

Yes, you are planning too much and have not spaced them out well.
July 3, I would not be able to handle the V and A Museum in a jet lagged state upon arrival. Go for a walk along the Thames or to St. James Park. Walk through Harrod’s.
July 4- Get tickets in advance now for both Westminster Abbey and the Churchill War Rooms.!
You will probably still stand in line but not so long. When we went to the War Rooms there was a very, very long line to the end of the block. We were moved to the front of the line when we showed the guard our tickets. Same for Westminster Abbey.
Your plan for July 4 is exhausting! Visiting Westminster Abbey and the Churchill War Rooms takes a lot of time, emotion and focus. When we left the War Rooms we were emotionally drained. These two sites are “must see” but you then plan to visit the National Gallery or Parliament followed by a musical. Way, way too much! After Westminster Abbey and War Rooms your brains will be mush!
The next day only an hour and twenty minutes for the British Museum? Or are you just going to visit one gallery, such as to just see The Rosetta Stone? Nothing wrong with that , just checking.
Fortnum and Mason has a very nice tea. You have a choice of sweet or savory foods. You each can chose a different one. Use this as your lunch one day. A lovely way to relax and be pampered.
Luckily you have a free day.! Move some of the places listed for first three days to the open day. They are all worthwhile but grouped too close together as your itinerary presently lists them.
Personally, I have to go to Westminster Abbey every time I visit London. My young grandchild called it a church that is a history museum, so true.

Your trip will go by in a blur-Changing of the Guard, British Museum, St. Paul's, Tower of London all in the same day? You need to slow down and enjoy your trip. It is best to plan one activity in the morning and one in the afternoon. At this pace, even if you manage it, you will be too exhausted(if not physically, mentally) to enjoy anything. I would cut the National Gallery/Parliament or Victoria and Albert, and St. Paul's. Others will disagree, but we like to have a relaxing tea the day we arrive. I have had the teas at both Harrod's and Fortnum and Mason and prefer FM. It's ok to wear business casual clothes there. It is not far from the National Gallery, although I don't know if you'll be awake enough to enjoy both if you didn't sleep on the plane. And you will have to be mentally alert to learn the use of the Tube-there are so many different lines, which fortunately are colorcoded on the maps.

You will need way more time at the British Museum and Tower of London. Nix the National Gallery and/or Victoria and Albert museum if you have to. Get tickets ahead for Westminster Abbey and the Churchill War Rooms.

Few thoughts:

Are you flying from the US?

If yes I’d consider walking outside for a few hours to fend off jet lag. From your hotel head to the river and go left heading towards Elizabeths Tower and Big Ben. So what if you have tickets for the next day.
Nice stroll along the Thames is an excellent way to adjust to the sights, sounds and smells.

Harrods closes at 6pm on Sunday.

Have you pre booked your Churchill War Rooms tickets? If not, do so. Very popular attraction.

https://www.parliament.uk/about/how/occasions/calendar/

Check this link for theatre options:
https://www.whatsonstage.com/

Wicked, The Jersey Boys, Lion King, Hamilton all good.

Pub grub is fine. And many pubs have gorgeous architecture, like the art deco Blackfriar.

For your open Wednesday visit Richmond. Nice neighborhood with nearby and expansive Richmond Park. Theres also Kew Gardens. The White Cross pub adjacent to the river is nice. Enjoyable to watch the crew teams practicing.

Have to get tickets for the Highgate Cemetary. Its a bit of a surprise to find Karl Marx grave there.

Go to Spitalfields. Or Brick Lane. Check the Tower Bridge lift schedule.

Great city. Lots to explore.

*EDIT: * forgot to mention if seeing the changing of the Guard is a must you can watch some of participants march back from the palace to the Wellington Baracks by standing along Birdcage Walk at Spur Road. ALSO investigate the 2 for 1 Days Out offering. https://www.daysoutguide.co.uk/2for1-london

Great City. Have fun.

I have no idea what time you'd need to be at the Tower of London to be sure of catching the last Beefeater tour at 3:30 PM, but I bet someone here does. However, tickets for the Tower are extremely expensive (about £30 for adults), so I'd want to allow time to see everything there of interest to me--which would include the crown jewels. Folks here have recommended making the Tower your first stop of the day, being in line before it opens and going straight to the crown jewels so you can appreciate them in uncrowded conditions. I assume prepurchasing an entry ticket for 9 AM needs to be part of that strategy.

The verger tours at the Abbey are worth the time and money. We arrived at Westminster Abbey last week around 9:30. 5 minutes in line. Next available Verger tour was 10:30 and lasted 90 minutes. Your allowance of 1.5 hours is not enough.

We found 2 major sites a day was plenty in the schedule.

I admire your ambition, but wow yeah way too much especially Tuesday. Westminster, War Rooms and National Gallery in one day is a lot, but you could push through. If anything, I’d skip the theatre. After a long day (and then dinner) it may be challenging to sit through. And sure it’s a very London thing to do, but is it what you really want to do? After a day of aggressive sightseeing, you should reward yourself with a relaxing evening. As essentially first time visitors this might be a good night for a tourist bus ride.

British Museum and the Tower are maybe your two biggest sights and require time and planning. Avoiding crowds should be a primary goal. One of these should be moved to Wednesday. I’m thinking Tower to Wednesday morning and see St. Paul’s after. If you’re still feeling game you can cross Millennium bridge to Bankside from there. Don’t worry so much about filling “dead space” in your day. Wandering a neighborhood in London is sightseeing too, there’s plenty to discover.

The tube will connect everything you want pretty quickly, but it’s important to remember that it still takes time. If you’re transferring between lines, you may be walking a good stretch underground. It can add up.

I spent half a day at the Tower. British museum could take literal days, but again in half a day you can see your personal highlights. 1 hr 20 minutes seems like a waste as it is a little time consuming to get to.

My mom and I spent a full morning at the tower, walked tower Bridge and all the way back down the Southside to Big Ben/London eye in an afternoon.

Look at a map and regroup your sites! I have zero issue stepping off a plane and going into heavy siteseeing mode, but London works well doing historical site/gallery + park per day. Repeat the next.

Thanks to everyone for their replies. I knew I had too much. I have rearranged for 2 items a day:
Monday: Westminster Abbey, Churchill War Rooms
Tuesday: Fun London Walking Tour for Changing of the Guard, The British Museum
Wednesday: Tower Of London, St. Paul's Cathedral
Thursday morning train to Edinburgh, Scotland

Still debating whether to do Victoria & Albert Museum on Sunday (we arrive early AM, Hotel check in 2PM) or what we should do that day. Also haven't committed to going to the theatre one evening. Open to suggestions & thoughts.

A lot of times, if the hotel wasn't full the night before and the room is ready, you can check in early, refresh, drop bags, and head back out. You schedule now is much more doable. You also have to remember it will be hot in July and that can take a lot out of you. Does your hotel have AC?

mitch, your new itinerary looks great.

you can check in early, refresh, drop bags, and head back out

Most places will let you leave your bags if your room isn't ready, then when you return you officially check-in ... we pack anything we might need on arrival day in our small backpacks so we're not rummaging through everything before we store the bags ... we've done this many times so we can keep ourselves going on the arrival day which helps with the jet lag and let's us get acclimated to our new adventure.... you should have time on your arrival day to do the V&A Museum and maybe even Harrod's if you're not too tired.

Sunday early AM - get Oyster cards and a SIM card for the phone and checkin or at least drop luggage at hotel. Go to Victoria & Albert Museum. Late afternoon/early evening go to Harrod's. Should we try to get some other sight in?

Sunday trading laws mean that all large shops have reduced opening hours (only a maximum of 6 hours) on Sundays. Your late afternoon / early evening at Harrods will need to move up. As Claudia said, they close at 6, and will be heaving folks out before that.

Those same Sunday trading hours will limit your ability to get your SIM early in the morning. Which shop had you in mind?

I contacted the hotel and they do have a luggage room where we can stash our bags.

I did not have a place in mind to get the Sim card. From others experience, is getting the Sim card for our time in London & Scotland a good idea? I know I should only use WiFi while there so there will be no roaming. Is there any easy way to turn off all apps that use Push notifications or do I have to get through the apps one by one and turn Push to Pull?
Someone suggested just downloading maps ahead of time. But while there, if someone suggests a place we should go (restaurant, etc.) I want to have access to maps to get me there. No car, just mas transit and walking.

All advice on the phone situation is appreciated. AT&T wants $10 per day, which isn't horrible. But I'm not sure if it's necessary.

Definitely go to the theatre one night! We just returned and saw 4 musicals in 10 days and thoroughly enjoyed them all.

Wicked is phenomenal. If I had time for only 1, that would be my choice.

Someone suggested just downloading maps ahead of time. But while there, if someone suggests a place we should go (restaurant, etc.) I want to have access to maps to get me there

I like maps.me ... download the app and map(s) of the area(s) you will be in and it works offline so you don't need any cell or internet connectivity ... very similar to google maps, enter an address and it shows you the way ... just be sure to download your maps when you are connected to wi-fi.

A suggestion for a show- Six. It’s about the six wives of Henry VIII and is absolutely fabulous. We saw the pre-Broadway run in Cambridge , MA (USA) and loved it.

I remember seeing info about Six on CBS Sunday Morning. I don't recall seeing it on the list of shows in London this July. I will loo again. Thank you!

Come from Away is brilliant! Phoenix Theatre.

Don't bother with an Oyster card for your few days there. Just use a credit card to tap and go. The TfL site tells you all about it. Or search the Travel Forum; there has been a lot of discussion.

You might consider a ride on the Thames Clipper (1 hour from Westminster to Greenwich) pleasant and amusing. And being outside will help keep you awake.

After a while, I found I liked riding the busses better than the tube: no stairs and windows that open and the difference in total travel time was negligible. The trains were stuffy sometimes.