First trip to England and Scotland, help.

I would skip Bath and Cotswolds-they are not located in your primary areas of interest. Allow 2 full days in York or maybe one more for a day trip. Don't get a car until you need it for York day trip. If you don't need car in York, take train to Edinburgh and get car for St. Andrews there. The more you utilize trains instead of a car, the easier your trip will be(unless you have a lot of luggage). You may find driving on the opposite side of the road more challenging than you expect-be sure to reserve an automatic transmission car.

When is this trip?

Ever driven on the opposite side of the road?

Have you calculated mileage for where you want to visit?

In Edinburgh, London, Bath and York, a car will be a liability. You won't use it to travel around these places once there, and it will be sat in an expensive car park for your stay. To travel between these places, use the train. To give you an idea, it is 4.5 hours from London to Edinburgh, and trains stop at York roughly half way. Best prices called Advance tickets, on sale about 12 weeks out and are for a specific train. If you turn up at the station on the day of departure for any train, it will be very expensive.
The centres of York and Bath are easily walkable, whereas London and Edinburgh are larger but have excellent public transport services.
You would need a car in the Cotswolds though, as it is a rural area.
Windsor is easily done as a day trip from London by train, it is only 30 miles or so from the centre.
Geographically, it would make sense to fly into and spend some time in London, with a day trip to Windsor. Train to Bath, then hire a car for exploring the Cotswolds. return the car. Catch the train from there to Edinburgh, I can see prices for £124 for the two of you on a random day in May.
Spend your time in Edinburgh/St Andrews then catch the train to York, prices as low as £46 in May. After York, take the train to Manchester Airport for as little as £17 in May. I know this deviates from your plans but it is the number 2 airport in the country and has many links.
If you spend more than £90 on trains, it would be worth buying a two together railcard for £30 that would give a third off prices. Prices above have that discount applied. When booking your tickets in advance, tick the box to indicate you have a railcard and purchase one from the station before you travel, you just need to have one when travelling, not when booking. You'll need a photo of each of you on your phones for this.
You could use air b and b if you want to, but there are many hotel chains in the country. A good one is premier Inn, a budget chain but very clean and comfortable and are all over the place, breakfast is usually £9.99 each/day. Another is Holiday Inn Express that has a breakfast included.

This would be first two weeks of April, 2025

I agree that if you are mainly interested in seeing cities then there is no point doing it by car. A car is essential for exploring rural areas easily.

Something to keep in mind is to be precise in your wording.
You say you want to see London "with the usual must see Tower, Westminster..."

I assume you mean Westminster Abbey, and if that's the case you need to say so.
Westminster is an area of town. Westminster Abbey is a church. Palace of Westminster contains the House of Parliament. Westminster Bridge cross the River Thames near The Palace of Westminster.

There are other references that need precision, as well, or you might end up somewhere miles away from where you intended. If you are in London and look up directions to Liverpool when actually wanting a short ride on the Tube to Liverpool Street Station, you'll end up in an entirely different part of the country.

Have fun planning and looking forward to your big adventure.

You could also take the train to Bath Spa(again the need for a precise name) and take a Mad Max Tour from Bath to get a taste of the Cotswolds.