Where to rent a car?

As long as you stay out of the very heart of the old center of Edinburgh, the driving near the city should not be difficult (and AFAIK there are no car rental agencies right in the center of things anyway, so it's probably moot). Pick up your car when/where you need it. Return it when you're done with it. Personally, I think it would not be a smart choice to drive from Edinburgh to London (unless that's how you want to spend your trip, which might be a reasonable choice, but it doesn't sound like that's your plan). There are probably many places you could pick up (or drop off) the car. The most obvious option would be at the airport - which is easily connected to the center of Edinburgh by train and bus.

To get from Edinburgh to London, there are two reasonable choices: fly (fast, and not expensive if booked well in advance), or train (may or may not be more expensive depending on when you book, and more comfortable but not quite as fast as a flight).

And...do be careful driving, especially the first hour when you make the big adjustment to "opposite side" driving.

We’ve got a trip coming up this fall, starting in Edinburgh without a car, and ending at the Edinburgh airport, when flying out of Scotland. But we’ll need a car in between, and have found some rental outfits in the center of the city, with surprisingly low rates, but not open every day, and short hours on weekends. So your exact schedule could influence from whom, and where, you’d choose to rent.

On previous trip, we rented from the Arnold Clark company, picking up in downtown Inverness, and dropping off at the very convenient Edinburgh airport. As David notes, airports are often a good place to pick up and/or return a rental car, away from inner-city chaos and traffic. Sometimes there’s an extra fee for using airport offices, but usually well worth it for the convenience. Check what companies charge, too, for returning the car to a different office than where you picked it up, but, again, those extra fees are usually minimal, compared with not having the option to take the car all the way from Edinburgh to close to London. Say you turned in your car in Oxford, if they have an office there, downtown or at the train station. Then just take the train, with your luggage, into London.

You might also consider taking the train a short distance (in the direction you plan to go anyway) and rent a car in a smaller city. We did that for dropping off our car; rather than drive even to the Edinburgh airport, we dropped the car in Perth and took the train into Edinburgh. It worked great for us and was definitely less stress than driving any where near the larger city.