In case you haven't seen it on this RS website, there's a section called Explore Europe. It includes many countries with lists of must-sees. England and Scotland are two of them. Click on the links for guidance.
The problem with having so little time is that you must do some serious prioritization. I was last in GB in 2016. I did a self-planned 6-week trip all by bus, train, ferry and one short flight. I arrived at London Heathrow (LHR), made a big circle starting in Bath and ending in London for a week before flying home from LHR.
It sounds like you have an idea of what ypu want to see in London. If this is your first trip to Great Britain, you could spend all your time there with or without some daytrips to other places. There is no shortage of daytrip options.
I had been to both England and Scotland twice before. This is a list of my must-sees and dos for the 2016 trip outside of London. They are not in any particular order and I saw much more than this, but they were my reason for going to the towns, cities or areas they're in. The many churches are primarily for the styles of architecture.
•Bath Roman ruins
•Bletchley Park
•Cotswolds
•Hadrian's Wall
•Durham Cathedral
•York and Yorkminster
•Hairy coos
•Stirling Castle
•Edinburgh Castle
•Royal Yacht Britannia
•Skara Brae and other neolithic and Viking sites on Orkney
•St. Magnus Cathedral on Orkney
•Costco near Aberdeen (not kidding)
•St. Machar's Cathedral in Aberdeen
•Canterbury Cathedral
•Hastings, the town and the battlefield
•Winchester Cathedral
•Evensong services in as many churches as possible
Please be aware that it's likely to be chilly and wet at the time you are going.
And if this is your first trip to Europe or the first one on a long time, thoroughly read and digest all the info covered in Travel Tips on this RS website.