Wonderful, three months is great. We love the British countryside which is very scenic and historical. We have traveled quite a bit in England, Wales and Scotland.
We did a great four week drive tour of England and South Wales, with no time in London or large cities.
Here is my detailed review of our trip with lodging, dining and tour information.
28 days in Britain and Celebrity Eclipse home
We spent the night at a farm near Haltwhistle to visit Hadrian's Wall, the name of the B&B is included in the link. Also, we spent three nights in Keswick. Not sure why you would spend four days in Haltwhistle?
I suggest that you do research for everywhere you visit and at least have an outline for what you plan to see. Consider renting a car, at least for some areas. Driving on the left is doable, but just be careful and take it easy.
Travel between each place, driving or taking the train or bus will probably take longer than you might think. For our four weeks, we only drove more than two hours on three days.
For England, I recommend Cambridge and Canterbury on the east side of London. Salisbury, Winchester, Windsor Castle and Stonehenge on the SW side. We haven't done Cornwall, but had a trip planned for that area, but cancelled due to COVID. We had planned to stay in Truro. You need at least a week there.
Bath and the Cotswolds, we loved, we stayed in Bath three nights and Chipping Campden for 6 (we used it as a base to visit the Cotswolds, Oxford, Blenheim Palace and Stratford Upon Avon). Don't miss Stratford Upon Avon, the Shakespeare history is special.
South Wales was special, we spend two days in Cardiff and two in Tenby. Also, we visited St. David's at the SW tip of Wales. We had visited North Wales earlier and loved Conwy. You might wish to visit Chester in England, not far from Conwy and Liverpool.
York and Yorkshire is special. We did three nights in York and also went through the Yorkshire Moors. Highly recommend Whitby.
Durham is worth a couple of days.
As far as Scotland is concerned, Edinburgh and the area north of the city is worth several day. St. Andrews is a must see. Also, don't miss the church where Robert the Bruce's tomb is located. Inverness in the north is great. We haven't done the highlands, but that is a must.
One thing, being gone for so long you will need to wash clothes several times. In the USA we call them Laundromats, but in the Uk they are called Launderettes. There you can get change and even have someone wash your clothes for you for a fee.
Since you plan to stay into November, I suggest that you consider starting your tour in Scotland and moving south. We did our four weeks in October and toward the end of the month we were in the Lake District and found it very chilly discouraging being outside a lot.