First-time trip to England/Scotland for 2 weeks. Please advise.

On arrival try to keep awake as long as possible on your first day to counteract jet lag. Walking in the fresh air is a good plan and you can see some sights.
There is no train to Bristol airport. Whilst you can go to Bristol for the airport bus it will be easier to take the direct bus from Bath.

This is details of the bus from Bath to Bristol Airport- https://www.airdecker.com/tickets/book-ticket/

I am sure people will know more than me about Lord of the Rings [assuming that is what is meant by Lot R), but it was filmed entirely in New Zealand. According to that source of doubtful knowledge, Wikipedia, you are looking at a place called Sarehole Mill (Birmingham) and rural Worcestershire (in lieu of the Cotswolds).
The Mill is accessed by buses from central Birmingham or a local suburban railway station called Hall Green.

For Jane Austen, as well as Bath, you are looking at some quite unfashionable places on this forum- Reading Abbey (especially the Gateway)- a quick and easy stop on your way to Bath or Oxford; Basingstoke and Steventon. Basingstoke is an easy stop on the other way to Bath (or by direct bus from Heathrow). Steventon is a 7 mile taxi ride away.
Also of course, Chawton. That is an easy day trip by train from London.
An hourly bus links Basingstoke with Alton in 49 minutes,
These are all real places associated with her, not movie locations.

For Harry Potter in London you have Kings Cross Station, Leadenhall Market, Big Ben, Harrow School and the London Zoo Reptile House. Picket Post Close at Bracknell (his 'home') is also an easy win if you go by train to Reading. Or on arrival day go there direct from Heathrow Terminal 5 on the bus then the train into London from Bracknell station.
Or Bracknell from Heathtrow, then bus 4 or 4a to Reading (for the Abbey as above) then train to London.

York will be busy and expensive, being a Bank Holiday weekend, but then it is busy and expensive at any time, so the busyness and expense is all relative. The expense is also relative to Edinburgh and London. Compared to those York is fairly reasonably priced.

For a first time visitor you seem to be planning to spend a lot of time travelling. You could easily spend 2 weeks in London with a few days out (eg Hampton Court, Windsor) plus an overnight in York. It's up to you what you want to see, but I notice that you seem to be planning to spend less time in London than in Edinburgh (and part of that time is when you may well be jet-lagged). Much as I like Scotland, there's a lot more to see in London.

Clearly it depends what you want to do with your time and what your key interests are but I would try to spend more time actually sight-seeing and less time travelling.

Regarding the Bank Holiday: that day and the rest of that week will be the Spring half term holiday for schools, so things which attract a lot of children eg the Natural History Museum in London will probably be busy each day and, as an indoor activity, very busy if its wet.

Have you got accommodation booked yet? May is a popular time to start going on holiday and English schools will be on half term holiday in the last week of May and accommodation may already be fully booked.

Your itinerary looks workable, although it does cover a lot of ground. For someone preferring 'to avoid crowds and overrated tourist spots' you are hitting all the main tourists spots. Be prepared for them to be busy, especially places like Bath and Bourton on the Water. Edinburgh and York are busy throughout the year.

You could take a Rabbies day tour to the Highlands from Edinburgh on the full day that you are in Edinburgh. I would take train from Edinburgh to York in late afternoon to give yourself more time in Edinburgh. I know you said you don't want to go to studios, but the Harry Potter one is really good and you need to make reservations far in advance. Unless you book (London is expensive)a room for the night before you arrive, you will most likely not be able to get into your room in London on your arrival. I would cut the Cotswolds as a separate destination, add a day to Bath, and take a Mad Max day Tour that includes a Cotswold visit from Bath. You could also add a day to York and take a day tour to the Dales, if you're a fan of All Creatures Great and Small. If you stayed in London longer, you could take a boat from the Westminster Pier to Kew Gardens.

I believe you'll find a bus tour of the Highlands originating in Edinburgh involves many hours of sitting in a van/bus and comparatively little time with your feet on the ground. Personally, I wouldn't do such a rushed trip to Scotland when it means so little exposure to the countryside--and really barely enough time for Edinburgh if you didn't set foot outside the city.

There are so many wonderful places to see in England that don't involve a flight.

London hotel rates for May are very high; I was unpleasantly surprised when I made my own reservations last week. Start looking immediately.

If you're interested in filming locations from the BBC Pride and Prejudice miniseries, the town of Lacock which stood in for Mereton is accessible from Bath by public transport, just about an hour away. It's a very charming and characterful place which feels a bit like stepping back in time.

Looks like you have no real time scheduled for London, it's massive and not doable in 1 day here and there. Yes, I would take out Edinburgh completely. It's not close. WIndsor's right near Heathrow, sometimes one doesn't get sleep on the plane and nap is required to stay safe. You can always add things once you're there, but scheduling some rest and seriously mapping your visit will save you lots of time and stress.
Happy Travels are important :)

London Walks does a Harry Potter tour of film sites in London. I took the tour a couple years ago an enjoyed it. They also offer a wide variety of walking tours in London. I've taken several and highly recommend them. I learned about them here on the forum.

Check them out!
https://www.walks.com/

From York you could take a bus to Pickering and then the North Yorkshire Moors Railway. The station is the one used for Hogwarts Station and the pub in Goathland (on the train route) is the one used for Hogsmeade. We were not looking for a Harry Potter experience but it was really fun discovering these.

Katie

We flew from London to Inverness and the next day took an all day Isle of Skye tour with WOW Scotland Tours. The weather was great and this tour was the highlight of our trip to England/Scotland. The day after the tour we took a train to Edinburgh, saw the major sights and then took a late evening flight to London. We could have spent another day in Edinburgh, but instead took a walking tour the next day of Oxford and the Cotswolds with London Walks. This walking tour was another highlight of our short visit.