Pick up car near London, drop off in Edinburgh?

So...would it make sense to arrive in London, take a train to Bath or something, and then rent a car to make our way around England and up into Wales and Scotland, and then take a train back to London

Yes, that sounds like a good plan. It is what a lot of people do.
The website for the train company from Edinburgh to London is here: https://www.virgintrainseastcoast.com/
Edinburgh to London Kings Cross takes about 4h20 to 4h40 (driving takes twice that) and there are two departures per hour (at xx:00 and xx:30). If you buy in advance (2-3 months) at the above website the prices start at £40, hopefully that won't kill you.

I strongly recommend an overnight in York. If you aren't visiting it on the way up north, it makes a convenient stop on the train trip south, as it is half way between Edinburgh and London and the trains stop there. There is enough to fill 2 days.

think about flying into one airport and back home from another - that saves you backtracking to your starting point

If you are heading as far north as Loch Ness, you can take a very long day trip to the Orkneys. The bus leaves Inverness early morning, heads to the John o'Groats ferry, then you get a smaller bus that drives you around the biggest island. You arrive back in Inverness around 8 in the evening. When we did that tour, stops included Skara Brae, Ring of Brodgar, and a few other spots. Our next trip we spent two nights in the Orkneys and saw a lot more, but if you just want a taste, the day trip is fun and very scenic.

For your trip, I would suggest if possible flying into either Edinburgh or Glasgow, rent a car, do a loop of Glencoe, Skye, the great Glen (Loch Ness), visit Clava Cairns and Culloden if interested. Drive south through the Borders, then visit Durham, Alnwick, Fountains or Rivleux Abbey, Return you car at York. Visit York. Train to London for a few days.

If you already have purchased round trip flights for London, then immediately take the train to York. Visit there, get a rental car, head up to Scotland and visit the areas listed in the previous paragraph. Return the car in Edinburgh. You could fly back to London or take the train, whichever is most convenient, and finish with a few days there.

If you haven't booked your flights it would make more cents and dollars to fly into London and home from somewhere in Scotland - perhaps Edinburgh or Glasgow. Save the expense and time of returning to London. And it helps to develop the attitude - "We will see that next time." --- so you don't feel so bad if you skip a place.

Thanks, everyone! Advice being factored in.... unfortunately we do already have our plane tickets, so we’re stuck with that. Any advice on Oxford? We’re definitely going there because I have a friend to visit. I’m thinking of making that our home base for several days, since lots seems to be within an hour or so of Oxford....

My wife and I planned four weeks in Britain, and wanted to visit places in South Wales as well as England and Scotland of great interest, as well as places where our ancestors came from.

After much planning we decided to skip going to London, since we had done that city already. Then we decided to skip Scotland, since we had visited Scotland twice on cruises and seen Edinburgh, Loch Ness and other places like St. Andrews.

We drove over 1600 miles and the trip was wonderful.

First tip, don't drive anywhere near London if you can avoid it. The SE of England is overloaded with traffic and you will spend more time wasted on the highways.

South Wales, the Cotswolds, Yorkshire, Durham area and the Lake District (NW England) don't have as much traffic and you will not get as frustrated.

Second, be sure to rent a car with a good GPS or have a smart phone or GPS where you can find your way. Also, rent an automatic.

Third, Parking is hard to find in many cities and can be expensive. Plan your trip so your hotel or B&B is walking distance from the city or town and your hotel has parking.

Fourth, in planning your trip, plan on it taking longer than MapQuest says.

It is a long way to Edinburgh, consider skipping Scotland for another trip.

Here is a detailed writeup of our trip on cruise critic, since we took a transatlantic cruise back to the USA after our drive tour.
28 days in Britain and Celebrity Eclipse home
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139

Hello,
You might consider turning in the car at Inverness and using Easy Jet to fly back to London. They fly to Luton and Gatwick, I believe. Transportation into Victoria Station in London from Gatwick Airport is easy by the Gatwick Express.
If it were me, I would spend the first days in London. By the end of your time in London, you would be over jet lag and enjoy the car more. It’s not a good idea to rent a car straight away after a big time change. There are several locations on the outskirts of London to rent a car where traffic is not a problem. It is easy to take the Gatwick Express from Victoria Station to Gatwick airport and rent the car there. I’ve done that. Renting at an airport is more expensive sometimes, but the supply of cars is usually good. Driving out from Gatwick is not an issue. It is easy. If you have a GPS, take it with you. Otherwise, rent one. And be sure to have a good map so you know the names of towns along your route to plug into your GPS.
After London, i would drive around wherever I wanted, fly back to London, and head home from there. To me that elimates both the cost of trains (which is expensive) and loss of time.
I believe you said you already have you plane tickets. Otherwise, I would suggest multi-city tickets to prevent backtracking.
We’re going to have overlapping trips. My wife and I are spending July in England and Scotland. We’re flying into Edinburgh and out from Heathrow and driving all around in between.
Have a great trip.
Bioboy48