First Visit London!

Just 2 notes:
Highclere is not always open, be sure to check your dates.
You can walk the Abbey road area by yourself. There aren't really many Beatles sites in London, and the London Walks Beatles tour we took was our least favorite.

I lived in London for 20+ years, and still visit regularly.
Sheekeys and Rules are both great choices.
If you want another splurge, the newly reopened Simpsons in the Strand is getting good reviews and is now on my list.
Any time I'm away in a city, we always take the HOHO bus for a rest and for orientation.
If you need more WW11 experiences, the Imperial War Museum near Waterloo is a good bet, and possibly a day trip out to Beltchley Park.
The Henrietta looks good, too, but be sure to request a walk in shower if this matters to you - I can no longer safely climb in and out of a bath.

. But then I realized it really serves a purpose--so...maybe we take a London HOHO.

I would suggest you avoid these. They can’t use bus lanes and you will spend a long time sitting in traffic listening to out of synch recorded commentary which is mostly incorrect. If you want to go on a bus you can, you know, get on an ordinary bus.

You are not going to be able to walk to all attractions. For context London covers more than 600 square miles which is twice the size of New York. Hampton Court Palace is 16 miles from the Tower of London for example, but there are 272 tube stations and 330 railway stations plus 9000 buses so you will be getting around. Use contactless to pay for your journeys.

Much as I hate to say it, I think you'll miss Highclere Castle. It generally has a spring opening in April, and is not open to the public in March. This is fairly typical for stately homes. As an alternative you may want to visit Hampton Court Palace, which is an easier trip from London, far more historic, and will be open.

Aside from what you have scheduled, maybe a museum? As a wartime history buff, I'd recommend the Imperial War Museum. Also perhaps a market? I like Camden Passage.

Covent Garden is a good central spot, but you'll still need to get a tube or bus - London is huge. I cant think of anywhere in London that tourists are likely to go that is not safe.

How exciting for you, Jay! It's been a number of years since we've been to London but spent a week and had a great time; there's SO much to see!

Personally? You might reconsider the trip to Highclere and take a look at Windsor Castle or Hampton Court instead? Neither are off tourist radar, and I've yet to do Hampton Court (next time) but Windsor was well worth the trip due to its long, long history. St. George's chapel, especially, is a fave. You could also walk over to Eton while you're there: The Eton Walkway look like fun!
https://www.windsor.gov.uk/things-to-do/the-eton-walkway-p1405401

https://ae6ab00e-2aa3-4774-be50-c07df9554132.filesusr.com/ugd/241361_4da7de0c7cc0402fa9a04b66efa13f1f.pdf

And I suppose we'd like at some point the Underground/Tube experience
as well.

That's pretty much mandatory for London as, per Trelawney, the city is very spread out and doesn't really have a compact historic center, per se, like Rome or Florence. It's comprised of individual towns and villages that were absorbed into the smaller, original city over a loooong period of time: urban sprawl, don'tcha know. We didn't find the Tube system to be difficult to figure out.

Not exactly off tourist radar but less visited is the Treasures Gallery in the British Library (also called the Sir John Ritblat Gallery). Free, and an excellent choice on a rainy day (and even a dry one). Yes, you will need to take the Tube or other transport to get here. Rick gives it a high-five too!

https://classroom.ricksteves.com/videos/treasure-of-the-british-library

London also has many small museums. Two of my favourites- Sir John Soane's Museum, which is filled with an incredible collection assembled by architect Sir John Soane and preserved in it's original form by Act of Parliament; and Leighton House (which is in Holland Park), the home of Victorian Painter Frederic Leighton.

You might also like Spencer House, in St James's overlooking Green Park. Built as the London home of Lord and Lady Spencer, it is quite magnificent and not particularly well known- it's open only on Sundays by guided tour, and when we went, only 3 other people were on the tour.

HoHo buses can be a great way to get your bearings in many cities around the world but London is not one of them. The traffic means that you will crawl round the route, you won't see sights like Buckingham Palace as they are not allowed to go past, if you get off to see something it might take you a long while to get back on etc. There are a number of regular bus routes that you can take that go past a number of tourist sights and will only cost £1.75 per ride. I'm sure others can give you some specific route examples.

You will love London! It's one of my favorite cities in the world!

One thing you might want to do, instead of a HOHO bus, which can be pricey, is take one of the regular double-decker buses. There are several of them that go through major sightseeing areas, and you can just relax and enjoy the sights as you go along. I took two of my grandchildren to London last year, and they enjoyed that more than anything. They were 11 and 15 at the time and loved sitting on the top level of the bus, looking down at the street and checking out all the sightseeing activities. Here is a list that I created of some of the buses that will take you to prime sightseeing spots.

Bus Route 15: This double-decker bus routes you through the Tower of London, St Paul’s, the Strand and Trafalgar Sq. Great for sightseeing!
Bus Route 9: Another double-decker bus passes Somerset House, Trafalgar Sq, Piccadilly, Hyde Park Corner, Knightsbridge and Kensington High St.
Bus Route 24: This bus heads from Pimlico past Victoria to Westminster Abbey, Downing St, Trafalgar Sq and Hampstead Heath.

I would also second the recommendation for Hampton Court Palace. The history there is amazing. There are beautiful gardens on their grounds (and there should be flowers in March, as that is when we were there). Plus the castle itself is just incredible. There is so much history there, and it's a fascinating place to visit.

And I am putting in a big plug here for Westminster Abbey. To me, this is one of the iconic sites of London, and I think it's a don't-miss place. Not only is it beautiful, but the history in there is just amazing. You can find the tombs of so many famous people from the past, and the history of the building itself goes back centuries and centuries. It's just an amazing place to see. Even if you don't visit during the day, try and get there for Evensong, which they have every evening at 5 p.m. It's well worth seeing.

You'll quickly realize there's no way to fit in everything you want to see in a short 5 night trip. As you can see from the posts already, there is no shortage of suggestions! At least you have time to plan and prioritize and keep the leftover list for the next time (I've been three times and still have barely made a dent in my list).

Not yet mentioned, I recommend the CityMapper app to help you navigate public transit. It has very specific directions that are helpful to new users (like which exit to take to get where you're going efficiently). You'll each need a contactless payment method to use tube and bus (either a credit card or purchase an Oyster card). I also agree the HOHO is not a recommended activity in this busy traffic town. Mardee has given some great bus alternatives.

I also second Sir John Soane's Museum, it's quirky and bite sized and worth a visit. I love the cast courts in Victoria and Albert museum. The National Gallery is always a must for me. And there are a lot of free museums, this recent post has info (probably won't help with the decision making though, it's a long list): https://community.ricksteves.com/travel-forum/england/london-s-free-museums-definitive-list

We have plenty of theatre buffs in this forum so you should be hearing about your question on the venue. As it looks like that show is returning for a limited run, you'll want to purchase tickets ahead to make sure you see it.

My personal absolute favorite things I've done/seen in and around London are: Bletchley Park, Greenwich Observatory, Hampton Court Palace, Churchill War Rooms and visiting the shoe making facility at John Lobb. The aroma of the leather in the last is amazing.

And I am putting in a big plug here for Westminster Abbey

Ditto. IMHO, it's a not-to-be-missed for many reasons!

Even if you don't visit during the day, try and get there for Evensong, which they have every evening at 5 p.m.

Always check the service plan for the week on the Abbey website- but as a general rule Evensong does not happen on Wednesday evening (when it is replaced with said Evening Prayer, unless there is a festal day on the Thursday ). On major festivals on any day of the week Evensong can be replaced with a Eucharist service, preceded by said Evening Prayer at 4pm.

My daughter and I have made it a tradition to eat at Rules when we are in London. It's historic and pretty and the food is good. Usually the service is good too. (I'm a Downton Abbey fan, that's what got me there the first time!)

I agree with the others who say Highclere Castle will probably be closed to visitors in March, but check the website.

I highly recommend Westminster Abbey. I recommend getting there very early, be one of the first ones in the door. (book the earliest time) After you enter the Abbey, book a Verger Tour. While you're waiting for the Verger tour, you'll have time to look around before all the huge crowds enter. I got to see the side chapels and the Lady Chapel almost by myself. It was wonderful. And with the Verger tour you'll get to go into the Shrine of St. Edward the Confessor.

Churchill War Rooms will be an absolute must for you. Wife and I could have spent a lot more time there. We had to hightail for an entry at the National Gallery, but I could easily have spent 5 hours plus reviewing things.

Seconding Leighton House as a cool smaller museum. Part of one of my favorite days from our first trip to London.

The Imperial War Museum is very well done

The London Beatles Walk was my favourite of the walks I have taken with London Walks. It all depends on the guide and we had a wonderful guide who brought his own scrap book of pictures with himself and The Beatles.