8 days in England

I bought a book several years ago on Amazon and I see there are several updates since mine.
But, I love this book for info and ideas for day trips from London.

Frommer's Best Day Trips From London

You can find used copies on Amazon and they may have copies at your local library.

You can see Rick's favorite and most practical suggestions at https://www.ricksteves.com/europe/england/itinerary or presented a bit differently under At a Glance at https://www.ricksteves.com/europe/england. Although that link suggests that you could get by with less time in London (based on getting up early every day), I certainly agree with your plan to spend 4-5 days there. I encourage you to add the extra days, if possible. You won't regret it.

Most of these are accessible by train, except that Stonehenge requires either train+bus or bus tour excursion and the Cotswolds are also much easier to see with a car.

We also love history and especially English history. Britain is loaded with history. Since you have 3-4 days outside of London, you have many option. York is an idea, but that will involve some travel time for your short visit of England.

I would suggest two options:

1) stay in London and take day tours of Cambridge, Canterbury, Oxford, Blenheim Palace, Bath, Salisbury, Stonehenge and more. Some tours can take in multiples of those places, like Bath, Stonehenge and Oxford in one day.

2) To minimize travel on the day strips, you can stay overnight on a multi day tour or rent a car and do it yourself. You can go to the Cotswolds for three days and see Bath, Oxford, Blenheim Palace and Stratford on Avon as well as enjoy the Cotswolds.

I would suggest renting a car for the last days...it makes life easier; especially for a family. As mentioned the Cotswolds is much easier seen when you have a car to see all the charming corners. As you say you are interested in history and art look up the Treasures Houses of England. These are the Great Estates which have tons of history and fabulous art. Blenheim Palace is one example. Look at the National Trust UK site for many ideas. They have an Overseas Visitor Family Pass. I believe that they have a 7 day one. You could see the Free stuff like the British Museum and the National Gallery in London then use it for paid entry things like the Tower the rest of the time. Bon Voyage

Since you said you like history and art, while in London, I would recommend seeing
The British Museum
The V&A--Victoria and Albert Museum
The National Portrait Gallery
The Museum of London
The Churchill War Rooms (if a WW2 history buff)
The Tower of London
Westminster Abbey
and that's just a start!

For your "extra" days, you could continue to sleep in London, but do day trips out of town. One of the best companies that does day trips out of London is The London Walks company.
www.walks.com
You meet the group and your guide at a train station, hop on the train, train will take you someplace like Bath. The guide will give you a tour of Bath, then escort everyone onto the train and back to London.
They have day trips to Stonehenge, Oxford, Cambridge, and many other places. And their tours are not expensive.
The London Walks Company has several day trips, with a guide, out to country houses. For example, this trip out to Blenheim:
http://www.walks.com/our-walks/oxford-blenheim-palace
Take a look at their website for more trips.

There are many companies that do trips (out of London). Here are two good ones.
Golden Tours (Gray Line):
www.goldentours.com
Evan Evans Tours:
www.EvanEvansTours.com
Both of these tour companies pick up customers at hotels around London, or the Victoria Bus/Coach Station.
You travel on a bus (coach) to various locations, then back to London at the end of the day.
Golden Tours has a good one, in which you see Windsor Castle, Oxford, and Stonehenge in one day.
Another goes to Bath, Stonehenge, and Salisbury. Salisbury has a wonderful cathedral.
Another goes to Warwick Castle, Stratford-Upon-Avon, the Cotswolds and Oxford.
I think any one of these three trips would be fantastic.

This is one way not to have to move out of your hotel (or apartment) in London, yet travel and see other sights far from London.

I also recommend London Walks for their two-hour walking tours all over the city and covering many topics. We did the Street Art tour and the Jewish London tour. You don't have to RSVP; just show up at the landmark or street corner where the tour meets and pay the guide (each of our tours cost 10 pounds when we did it a year ago). The guides are very knowledgeable and they point out things along the way that may not have to do with the topic, but are interesting in their own right.

I did two day trips from London, York and Canterbury, and I recommend both. In York, I walked the city walls all the way around; went to York Minster, where they offer free tours of the cathedral; visited the Shambles, a medieval street; and went to Clifford's Tower.

In Canterbury, I visited the cathedral and the Norman castle. The castle is ruins now but I loved it. It's between a traffic circle and an apartment complex. How cool to live next door to a castle!

July is vacation time for UK residents as well as many Europeans, so take that into consideration in terms of crowds. This might argue in favor of some of the more out-of-the-way sights, both in London and on your out-of-London days. You can find history and art just about everywhere, so you might dare to bypass some of the ultra-famous sights and go for the quirky offbeat gems.

I'm going to add on those 2 days straight up.

Day 1 - Fly overnight to London
Day 2 - London sights
(Buckingham Palace, Houses of Parliament, Westminster Abbey, Trafalgar Sq, National Gallery)
Day 3 - Day to Hampton Court (& Kew Gdns??)
Day 4 - London sights
(Tower of London, St Paul's, Tate Modern)
Day 5 - Day to Windsor Castle
Day 6 - Thames to Greenwich, British Museum
Day 7 - To Salisbury (pick up car)
Day 8 - Via Stonehenge to Bath
Day 9 - To Cotswolds
Day 10 - Exploring Cotswolds
Day 11 - To Oxford
Day 12 - Return car and fly home from Heathrow

Your revised plan looks sensible other than the last bit - driving in Oxford is a
nightmare and parking difficult, so drop the car there on arrival and take the train back to London.

Agreed about the parking, especially in Oxford. We are planning a drive tour next year and will train into Oxford instead of driving.

Here's another vote for staying in London and sprinkling day trips in throughout your stay...Greenwich is at least half a day (take a boat at least one direction). Hampton Court is easy to get to by train and a full day with travel. Windsor is a day (I read you should take a boat one direction). Personally, I've done four Evan Evans day trips from London (including one to Paris) and it's a great way to see the area. The best was Windsor Castle/Bath/Stonehenge. Big time. Another to consider is Oxford/Cambridge. This summer I plan to take the train to Brighton for a day. No reason to move and change hotels with all the time and hassle that involves....unless it was to Edinburgh, a whole different conversation. But also, you could easily fill up the whole time and not leave London...depending on your interests, there's plenty of art (a vote for Courtauld Museum), military history (a vote for the Churchill Museum/WW2 War Rooms), British History (a vote for all the usual must-do's -- Tower of London, Westminster Abbey, St Paul's Cathedral, changing of the guard, and the under-the-radar-but-good Royal Mews)...and also the gondola ride that ends at Camden Market (which is a great site for unique stores and food) and the Fullers Brewery Tour -- free samples at the end! So much to choose from....Enjoy!

If your interest in history extends to war history, I suggest not only the Imperial War Museum (Lambeth North), the Guards Museum, the RAF Museum (Colindale in North London), and the National Army Museum. You could take a day trip to Duxford where the extensive RAF Museum is, or a day trip to the Bovington Tank Museum, both of which accessible by train from London.

David from Australia, as usual, has given you an excellent itinerary. He is very good at working out the perfect schedule.

I will just tweak his itinerary a little bit....picking it up with Day 7. ***Only to show how to do it without a car.

Here's what David wrote:
Day 7 - To Salisbury (pick up car)
Day 8 - Via Stonehenge to Bath
Day 9 - To Cotswolds
Day 10 - Exploring Cotswolds
Day 11 - To Oxford
Day 12 - Return car and fly home from Heathrow

Changes I would make:
Day 7--Train to Salisbury; Do not pick up a car.
Drop your bags at a hotel. The Stonehenge Bus leaves from the center of town, 1/2 block from the tourist information office. (Marked "TI" on most maps.) You can buy a ticket to Stonehenge at the TI office, that includes the bus ride out (very short) and back, plus admission to Stonehenge.
After you return to town from Stonehenge, see Salisbury Cathedral. They have an original copy of the Magna Carta on display. Then walk around the small town of Salisbury. Cobblestone streets in some places, great old buildings, good shops and cafes.
Spend the night in Salisbury.
I can recommend a couple of good hotels if you need that information.

Day 8--Travel to Bath by train or bus. It's not that far from Salisbury. Stay 2 nights in Bath.
The first day you arrive, drop your bags at a hotel and walk around and enjoy Bath. See the Roman Baths and take the walking city tour.

Day 9--The second day you are in Bath, use this most excellent small van tour company to see the Cotswolds:www.madmaxtours.co.uk
They do a great one day tour that goes to a bunch of Cotswold villages. Sit back and enjoy someone else doing the driving!
You will want to do your Cotswolds tour the second day, as they leave Bath fairly early in the morning.

Day 10--To Oxford by train or bus from Bath. You will be glad you don't have a car once you see the traffic in Oxford! Spend the day exploring around Oxford. Sleep one night in Oxford.

Day 11--Train back to London. (There are trains leaving for London at least every 30 minutes.) Spend this night in London. (Near Heathrow Airport, if you have an early flight home the next day.)

Day 12--Fly home.