We spent four weeks in Scotland and England and felt we had only scratched the surface. We had planned to go to Wales, also, but decided at the last minute that there was just too much to see. Let me know if you would like the itinerary we used and I'll be glad to post it. We plan to go back and spend at least two weeks in Northern Ireland and the Republic of Ireland.
Mentioned in this answer:
It's possible, of course, but will give you only the most cursory of experiences. Maps are deceptive.
England, Scotland, Northern Ireland and the Republic of Ireland (an entirely separate country with a different currency) ...and what about Wales?.....are all absolutely stuffed with historical sites and sights, beautiful countryside and fascinating places to visit, explore, walk, eat and drink.
You could 'tick boxes' in all of them in a 3-week trip but, to be frank, you'll be disappointed if you think you can do anything more than tick boxes. Much better, imo, to focus on exploring either the island of Ireland or the island of Great Britain (England, Wales, Scotland)...or England & Wales, or England & Scotland, or England and the Republic of Ireland. But even if you do any of those options you still won't see everything in just 3 weeks.
Mentioned in this answer:
Darren
Sandra