We did a 28 drive tour of Wales and England last October. It was great. We didn't go to London or the SE England or N. Wales, we had done that before.
Here is my detailed trip report as well as our cruise home to the USA.
28 days in Britain and Celebrity Eclipse home
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139
We have been to Scotland and Ireland. I think a week in Ireland is enough. Also, I think a week in Scotland is enough. The more you research and plan your trip, the more things that you will want to see.
If you rent a car, be sure to have a navigation system or gps or cellphone with google maps. Also, if not used to driving on the left, get an automatic. Further, plan your drives to take more time, especially in Southern England with all its heavy traffic. Plan on spending some time in Northern England, Yorkshire, Durham and the Lake District are special. Also, Wales is special. I recommend for S Wales there are many castles to see. Tenby is a great place to stay. Don't miss St. David's.
The Cotswolds you can do from a hub, we used Chipping Campden to visit Oxford, Blenheim Palace, Stratford Upon Avon as well as the other towns in the Cotswolds.
You can opt to use the train, but if you don't rent a car and drive through the countryside, you will miss a lot of spectacular views. Check out eh book The Back Roads of Britain.