I’d love to accommodate you but we can only put up 2 adults and one kid - sorry!
However, do pass any places by me that you’re looking at, as some places look good value but are in rubbish areas. May is a good time for Scotland - it’s our summer!
Your schedule to visit different parts of the UK plus Iceland seems a little ambitious to me if you want to really see stuff - how long are you going for? We are absolutely stuffed with architecture and historic sites etc: there is literally masses to see. I would choose very carefully and spend at least a couple of days in each place, preferably more. Forget Iceland unless you are very outdoorsy. It’s pretty dreich (rainy and grey) and you can get a similar experience in the Highlands since you’ll be in Scotland (drive from Edinburgh to Skye for example is about 6 hours with a stop on the way.)
London: the young, hip area is now Spitalfields/Shoreditch. Well worth a visit for your teens. We stay at the wonderful stayinspitalfields.com which for London, considering she can give you 3 double rooms in a listed building is great value.
Other places to see in London, I think, are the galleries in South Kensington (the V and A, Science Museum and Natural History Museum) and take a walk around Mayfair, then up to the British Museum if you want to revisit. I wouldn’t personally go to Covent Garden - extremely touristy now. Soho you could go to at a push, but Shoreditch has now taken over as the arty place.
I’d take a light railway to Greenwich and visit the fan museum https://www.thefanmuseum.org.uk/: really worth a visit for the museum and for Greenwich generally.
I like to walk around places like Cheyne Walk/Chelsea and go visit places like: https://www.nationaltrust.org.uk/carlyles-house
Another must is a visit - nice in May - to Holland Park and www.leightonhouse.co.uk/
Go to Trafalgar Square and walk around (you can then pop into Soho and go to www.gelupo.com/) and go and visit the National Gallery. Another must is to walk along the river and visit Tate Modern.
These are the things we do and keep going back to do.
So far as UK places: Bath is a must, York MInster is amazing though one could say the same about stopping off at Durham Cathedral - both cities are quite nice but unless you have loads of time I’d stick to Bath and Edinburgh.
In Edinbugh, try to stay near where we are in the New Town. Drive up straight to the Highlands through Perthshire - you can stop on the way several places for a bit but don’t bother staying the night. Skye is a must (again, can’t accommodate you, I’m afraid, as we only have space for 4 - sorry!) and if you want more Highlands take a boat over to Harris/Lewis. These are our favourite places. Skye, however, will occupy a good few days. You can get a train up to Skye, it is possible on public transport but hiring a car obviously gives you more flexibility.
I wouldn’t overnight on the train from London to Edinburgh personally. It’s a 4hr 40 min fast train during the day and about 11 hours during the night. But book in advance - 2 months preferably - for best prices and buy a ‘family/friends’ railcard: that makes it much cheaper.
That’s all I can think of for now!