As emma suggests, some compromising is necessary for any travel, unless one has unlimited time and finances! If you're determined to see a bit of England and Scotland it's certainly doable. Our first trip (2 of us) years ago included starting in London, taking a train to Edinburgh for a couple of days, then stopping in York for 3 nights on the way back to London. I'm not sure we even had 12 days total, but you go when you can and for how long you can. This past August, we'd initially thought about seeing more of England and Scotland on a 2 1/2 week trip, but decided we'd limit it to just locations in Scotland.
After flying from Denver to London Heathrow, we transferred immediately to another terminal to catch a flight to Edinburgh. If you're already at the airport, that saves getting into London to catch a train at some point, and the flight to Edinburgh was quick, comfortable, and well-priced. You could fly to Glasgow instead, but check the flight times and availability.
After a week in Edinburgh without having rented a car yet, we flew to Shetland, then ferried to Orkney. A flight on flybe Air to Kirkwall, Orkney is worth considering to maximize your time, but you could also drive if that met your needs. After arriving, getting the hang of deliberately driving On The Left took some time, and it was extremely helpful having a second person to help remind the driver to keep the car from going too far to the left. Except for the Motorway, the roads in Scotland all seemed narrower than we're used to in the USA. While the secondary highways are wide enough for cars and even big trucks, there's not a lot of excess elbow room, and you have to focus on staying inside your lane and not drifting off the left edge. We were there at the height of tourist season, and traffic was likely heavier than you'd experience in May. Not all the other drivers were adept at staying on their side of the center line, and in the countryside, you might be sharing the road with wandering sheep, too.
The UK has enough to occupy a traveler that you could easily fill every day with no actual agenda and would come across serendipitous finds. And you'd have a minimal, if any, language barrier :-) You may be well aware of this, but London has both the Tate Britain and Tate Modern museums.