Vet my 4-day London itinerary

Just for reference — I’m used to walking about 8 to 10 miles a day when touring and have been to London several times. Day 1 seems doable to me but I think you need to work in the Houses of Parliament on the day you visit Westminster Abbey because they are practically next to each other, Day 2 is too ambitious and out of sequence. You can easily spend four hours at the Tower because—after the formal tour—you’ll still need to see the Crown Jewels and the armour museum along with other sections the tour does not cover. After the Tower, you can see the Tower Bridge and cross over the Thames to walk along the southern bank. You can see the HMS Belfast cruiser ship that saw action during WW2, Shakespeare’s reconstructed Old Globe Theatre, and the Clink prison ( 1144 AD) on the way to the Tate Gallery. The Clink has a hamburger restaurant in it if you haven’t had lunch by the time you arrive there. If you’re able to stay on your feet for even a portion of the Tate’s collections—. you’ll be exhausted and need to call it a day. Save the Houses of Parliament for another day.
On Day 3 when you are going to Hampton Court Palace you might want to visit the nearby Kew Botanical Gardens which are seven miles from Hampton Court. Plan on three hours if you do visit Kew Gardens.
London is such an amazing city!

After the Tower, you can see the London Bridge and cross over the Thames to walk along the southern bank

I would cross over Tower Bridge from The Tower to the South Bank (which is the bridge often erroneously called London Bridge by tourists). London Bridge itself would be an unnecessary detour along the north side of the river.

Yes- I think I may have muddled the sequencing of the Tower Bridge.

I need more information...

1) When are you arriving in London and at what time?

2) Where are you traveling from?

3) What airport or train station?

4) Where are you staying?

I had the same thought as Amelia: The British Museum and the V&A are both very large. To see one of them comprehensively would take more than a full day. A tour is an option for seeing some of the highlights in a shorter period of time, but it wouldn't be my choice if you have special interests, because the tour might not touch on those at all. For example, I love glass, and the V&A has a nice collection. I've probably spent roughly two hours there on each visit, and I bet a museum tour wouldn't even step into that exhibition hall.

If you plan to see the British Museum just during the tour, not staying for further exploration, you might see whether there are tours that start later in the afternoon. The British Museum is unbelievably packed by museum standards. That applies especially to the ground floor, where a lot of the most popular exhibits are located. In my experience, the crowd thins out somewhat later in the afternoon. Museum staffers have told me it's worst on weekends/holidays, Fridays and rainy days. Perhaps you can put that information to good use. I'd also recommend that if you visit the museum independently and find it horribly crowded upon arrival (which is all too likely), you should start with things of interest to you on the upper floor, then move downstairs a bit later, when the crowds may have dissipated a little.

For non-tour visits to the large museums, it's highly advisable to take a look at the museum layout ahead of time. You'll be able to find it on the museum's website. That's the key to prioritizing your visit. I'd also recommend printing a copy of the floor plan to help you navigate. You can no longer be assured of finding printed maps available at museums--though I don't know about the current situation at the places on your list.

Several of the major museums in London (including the British Museum and V&A) are open late one night a week. That's a way to squeeze in a bit of extra sightseeing. If you intend to take advantage of the extended hours, stop at the information desk on the way in and ask whether all parts of the museum will remain open. In the past only the ground floor of the V&A remained open after the usual closing time. Also check on whether there will be any food available during the late hours. At the time of my most recent visit (a few years ago), the British Museum food options all closed before 5 PM--maybe as early as 3:30 PM. The security line for teh British Museum is significant (book your entry on line to minimize that); you will not want to exit the museum for food and brave that line for re-entry--though my observation is that the line is a lot shorter late in the afternoon than it is in the morning. I usually carry a little package of nuts with me when I head to a large museum. In a pinch I can head for the restroom and fortify myself with nuts.

I agree with the others particularly:

Day 3: Honestly, I'd start with Hampton Court. Take an early train and be there when they open. See St Pauls when you get back to London if you have the time and energy.

Day 4: I'd pick one of the other, then do a walk in the other half of the day. I particularly like British Museum but I pick and choose there and limit myself to about 3 hours max as it's huge, can be crowded and is a lot to take in.

Were you trying to work in guided tours when you say "tour"? Or did you just mean you'd see it thoroughly?

Looks like fun, albeit busy! On day 1, you're rushing it to have lunch and Trafalgar Sq, walk to Buckingham Palace and then try to walk past Big Ben before a tour of Westminster Abbey. It makes more sense to save that for your day 2 tour of the Houses of Parliament, since Big Ben is part of the Parliament site. And do your Westminster Abbey tour on the same day if you can.

Day 2 - most of your sites are in the east, but you're heading back to Westminster for your Parliament tour, so you're going to waste time traveling.

Thank you all for the insightful comments.
@Frank II: I don't have the answers to your questions yet, except for the second; I'll be coming from the US.

Here's my revised itinerary.

Day 1

9:30 AM - Westminster Abbey

? - Walk by Big Ben, Buckingham Palace.

? - Lunch at Wesley's Cafe/Westminster Arms

? PM - Houses of Parliament (arrive by 2 PM)

If time and energy permit

? - West End Walk
    Leicester Square, Covent Garden, Charing Cross Road, Denmark Street, Soho Square, Brewer Street, Picadilly Circus.

Day 2

8:30 AM - St Paul's Cathedral

11 AM - Tower of London

? PM - Lunch at Armouries Cafe

After the Tower, see the Tower Bridge and cross over the Thames to walk along the southern bank. See the HMS Belfast cruiser ship that saw action during WW2 on the way to the Tate Gallery. A coffee at the Tate Modern offers relaxing Thames views.

London Bridge, Borough Market.

Day 3

10 AM - National Gallery

?- Lunch at Trafalgar Square
    Café in the Crypt, The Chandos

? - British Museum

? - Dinner at Charlotte Place
    Salumeria
    Princess Louise Pub is nearby

Day 4

9 AM - Catch train from Waterloo for Hampton Court Palace

? - Lunch at Tiltyard Cafe

West End Show - Stranger Things

Oh, that looks like a much more enjoyable pace to me!

Plus you've done your research so if something goes sideways and you need to have a change of plan you know options on what you can see. Those items are also the start of your "To-Do" list for your NEXT trip to London, hahahaha!!

And yes, many of us have never-ending to-do lists for London (and Paris)!!

If any of the previous day's events spill over, I'll sacrifice Hampton Court Palace and stay in the city instead.

We all have different levels of interest and how much time we want to spend at a site. For me, Hampton Court was the highlight of our London trip and we spent 6 hours there. We also spent a full day at the Tower of London. The museum inside the White Tower is outstanding.

If Day 1 is your arrival day from the US, I wouldn't try to do half of what is on either of your itineraries but jet lag is not my best friend, YMMV. Depending on your exact dates, London Walks (walks.com) might have a scheduled outing that suits you. If you are leaving London to return to USA on Day 5, depending on the earliness of your flight you might have a bit of regret going to a late show the night before.

I’d check your dates, Parliament tours are usually only on a Saturday so I highly doubt you’ll be able to tour St Paul’s the next day as it would be a Sunday and the church will be used for worship services in the AM