I went to the UK on vacation two years ago, and the only problem I had with it was that I only spent one night in each place.
I started in Edinburgh, spent two days there, went up to Stirling castle, and then into the highlands for a few days, (Fort William, the Isle of Skye, Loch Ness, etc) went south and spent a day in a quaint village in the Scottish borderlands (If you're into iron/bronze age ruins and stuff, 40% of all of the hill forts in the British Isles are in the Scottish borderlands) From there I went to Hadrian's wall, and spent the night in Durham, went to York for the next two days, go and walk the old Roman walls if you can. From there I went southwest into Gwynedd (Northern Wales). Go check out the castles at Conwy and Caernarfon, pretty interesting, plus, the smallest house in the UK is in Conwy. I would suggest heading south into central Wales if you get the chance, I hear its quite scenic. After that, I went through the Cotswolds, an area of classic small British villages in between Bristol and Oxford, go to the local pub in one of these villages if you can, you'll get a good experience of culture from places like that. Stratford upon Avon is in Warwickshire I think, birthplace of Shakespeare, there's the Royal Shakespeare Theater, book tickets in advance if you're into his plays. From there, I went to Bath, obviously you need to go to the Roman bath complex there. After that, I went to Stonehenge, and also hit Avebury, a super small village inside an enormous henge that's a ton less crowded, tho watch out for the sheep among the stones. Oh, and you might want to join the National Trust to get discounts on a bunch of places like Hadrian's Wall, Stonehenge, and a bunch of other little things that you can get into for free with it. From there, I went to London and spent three days there. Go check out the Tower of London, the British History Museum, and Westminster Abbey for sure, and there's thousands of other options as well.
You can use apps stuff like AirB&B to find lodging, where you can rent out a room in some house for a night in places, which was very useful when I was on the Isle of Skye. That was my trip, but I excluded quite a bit, such as places up near Orkney, and the upper scottish highlands near Wick, Cornwall, much of the English coast, and basically all of southern and eastern England. Try to include as much as you can, but stick to places for two or more nights, otherwise its just too hectic.
If you really want a good experience of the British Isles, you're gonna have to do it in at the very least three 1 1/2 to 2 week trips. I'm heading to Ireland this summer, and I will hopefully return to go to northern scotland along with southern and central England. But for now, one trip will do. Remember, cities aren't everything, and in some cases, they hold the least that you want out of a vacation. And you don't have to see everything, so don't rush around all the time trying to catch a glimpse of every last thing. Its called a vacation for a reason, and its to relax, not to run around. Smaller and less-visited places can hold as much or even more charm than all of the big attractions that everyone goes to.
And I had Fish and Chips in a seaside village on the Firth of Forth, and they really do not taste like plain kerosene. If you find the right place to get them from, they're amazing.
Good luck on your travels!