London's accommodation for 10 days

Where: anywhere within or on the circle tube line (yellow on the tube map), or on the "south bank" of the Thames.
That gives you hundreds of paces to stay, all equally convenient as everywhere in that area is an easy walk to a tube station or bus stop.
Specifics, start with the "Premier Inn" chain. They are what is described in English as "cheap and cheerful", that is basic, no frills but good value: https://www.premierinn.com
Other hotels in that price range, or of course all prices above that.

• 9 days is tight if you are including travel days and if you still want to do everything in London. I would scrub the Cotswolds as unfortunately that is an area for touring by car, and you don't have time for that
• Bath is a possible day trip.
• Stonehenge is best combined with Salisbury. That is the nearest station, has a bus to Stonehenge and fortuitously has a cathedral worth visiting.

Bath + Salisbury/Stonehenge + Harry Potter is probably enough trips. But if you can squeeze in an overnight trip in, add York. 2 hours from London by train, walled city (walk the walls), city gates, huge cathedral, old streets, Roman and medieval bits. A good contrast to London.

Eight full days should be enough for the list you've included. You can do day tours of all four of the locations you mention which will give you time to see some of the sights in the city that maybe you didn't see last time AND see some shows in the evening.
Is your budget all-in or just for lodging? For three people at a hotel or B&B that's a bit low but if you choose an apartment through Airbnb or VRBO you should be able to find something in that price range.
Most of the places you list that you want to see are west/northwest of London so staying on that side of town will save you time getting out to those places. If you take public transport you will need to get out of the city and rent a car to see the Cotswolds and Stonehenge. Bath and the HP studio can easily be done just with public transport. Since your time is so short I would book a tour of the Cotswolds and Stonehenge and then see Bath and the HP studios on your own since they are easy via public transport.
Trains to Bath go from London Paddington and trains out to Watford for the HP studios go from Euston. Maybe stay somewhere in between...Marylebone or around Regent's Park.

Have a look at the London Walks website for their day trips to the Cotswolds, Stonehenge (and Salisbury) and Bath. Unlikely that their dates are up for May 2019 yet, but would be a good option if it fitted in with you plans. You get a Blue Badge guide and they have worked out the logistics. www.walks.com

I think you might benefit from an apartment over that length of time. Extra room, opportunity to do take away, simple meals. We like www.LondonConnection.com. Have stayed in their apartments 3 times now. Great staff, lovely apartments. It will be nearer the upper end of your range, but so will a hotel for the 3 of you. Also, it is terrific to have a clothes washer so you can pack light.

Something we do when we have a long stay in London is to pre-order groceries online from www.Waitrose.com and schedule them to be delivered a couple of hours after we expect to arrive at the apartment. Saves schlepping and ensures we have essentials on hand quite easily to get us rolling.

Note that many hotels (Premier Inn for example) charge extra for breakfast and on a per person basis for 9 nights it is steep. We can do that for 2 or 3 days, but after a while, the "full English" is a bit much and I'd rather have toast or yogurt!

You might want to consider splitting the time between London and Bath. The Cotswolds and Stonehenge are easy to get to from Bath.

Mad Max tours offers one day tours in small vans to these areas.

"Hub" hotels by Premier Inn are great. I stayed at the "Westminster Abbey" one last summer and the location is wonderful (1 min walk to tube station, 2 minutes to Westminster Abbey itself, very central". The hotel is brand new, everything very clean. About the cheapest place I've stayed in London. Breakfast buffet superb (but costs extra, still not much more than a coffee and croissant down the street and is a huge selection).

Perhaps go to Bath upon arrival for 3 days and do Stonehenge and the Cotswolds (hard without a car) and spend the rest of the time in London. But you could do them all as day tips from London as well and that way you could break up the London time with day trips.

Thank you everyone for all the suggestions. I will leave Cotswold for next time and concentrate just Bath and London ( thanks Chris).
My plan is after landing in Heathrow head straight to Bath for few days, visiting Stonehenge on the way back to London. The rest of the time we will stay in London.
I will check out all the recommendation for lodgings ( thank you Laurel and Isabel ). Airline has been booked and I will check out the Mad Max tour and the other tour as Frank and Katy have mentioned.

Trang, the only easy access to Stonehenge without a car is by bus from Salisbury. This is not on the direct line from London to Bath, but you can catch a train to Bath to Salisbury, then after visiting Stonehenge from Salisbury to London, arriving at a different station in London to that which you left for Bath from.
I don't think you can do both Bath and Salisbury/Stonehenge in one day, too much travel. You could do an overnight in Bath.

You can look up train times and buy tickets on the National Rail site: http://www.nationalrail.co.uk/

Thanks Chris for clarifying the travel route. I am planning to stay in Bath for 2 days. If it is not convenient for us to travel to Stonehenge from Bath then we will return to London and do the trip from London. There are more choices to get to Stonehenge and Salisbury from London I think !!!. Again thank you for the advice. 😊

A great way to travel from the airport in London to Bath is to use an outstanding private car service, russellsofbath.com. After a long flight, it was great to be met in the airport and driven directly to the accomodation. Danny, the owner, as well as his driver, Marc, provided us with excellent customer service - very prompt and easy communication, on time, safe driving, and nice cars. We used them for some day trips from Bath since some villages are not easy to get to by public transportation. His prices were very reasonable.

With less than two weeks you are making the right call to stick to one direction from London. You can also catch and direct bus to Bath, I have done it. I like staying in the area around Victoria Station in London (Pimlico/Belgravia), recommended in Rick’s book.

Thank you lalfordduran and vftravels for suggestion alternative mode of transportation from LHR to Bath, I will add this to my list of researching . When I travel on my own I usually taking bus or train but this time with my nieces (younger generation) they want more convenience so it will be a compromise between us when traveling.

hi trang
i'll let others tell you about train travel (book early for discount fare) or other transportation. when we were in london we took train to bath, walked around, saw baths, did the hop-on/hop-off thru bath which was good. after the ride we went to sally lunn's eating house for their famous bath buns and tea. caught the train back to london.
with you staying in london i would rent an apartment. space to move around, kitchen to cook in and have breakfast, buy groceries, have take-out meals, maybe a washer for your length of time, sit on a chair or couch instead of a bed.
you can check booking.com click your filters, cross-pollinate.com, homefromhome.co.uk among others mentioned here. read the fine print, look at pictures carefully (a bedroom is not a living and dining room with a sofabed, shared bathroom down the hall, cancellation do's and dont's, cash euros for payment or security deposit, stairs/steps or lift to climb with luggage, location is near train/bus) when you see what you like book early, you get pick of the crop.
some things to interest you:
b-bakery.com afternoon tea tour on the big red bus.
jasons.co.uk london barge canal ride.
eatingeurope.com food tours in london
visitlondon.com has lots of things to see and do, maybe matinee tickets at the theatre.
streetsensation.co.uk map of street markets to look or buy plus street food.
of course convenience is the "younger generation" so compromise is let them pay the transportation costs and you can sit in back sit and enjoy. that's what my nieces do for me since i've taken care of the rest. let us know how you do with the4 planning. you and the "gang' will have a great time.
aloha