Return hire car in London

drive it round London

No - you don't want that - the tube (or walking) is faster and a bus is much more relaxing.

Shed it outside London and take the train to your hotel. Probably the best place to shed it is Heathrow airport but it depends on which company you use.

If you decide to keep the car, be aware of the extremely expensive Congestion Charge for driving in central London, the extremely expensive parking - very few places have their own parking and you can't leave it on the street - and the extremely slow and heavy traffic in London. If your car isn't quite new - probably it will be, but I don't know where you are getting your car - you will also have the extremely expensive ULEZ Charge (Ultra Low Emission Zone).

I don't know how many in your family but I do know you would need a large family for Oyster charges on the Tube to get close to parking charges let alone all the other costs of having the car. Your stress level might enter into it.

If you want to drive in the vicinity of the British Museum you should know that some previously fairly wide roads nearby are being changed to narrow two-way roads with special bus stops and much larger pedestrian areas. The construction of these changes is playing merry hell with the driving around there at the moment.

There's a lot of construction areas around the Cross Rail stations.

We are now only allowed to state facts and not give advice, lest we be seen as bullying, but a fact is that I wouldn't do it. And I have to drive into London often.

You cannot drive around central London. All the places you want to see will be either on streets you can't stop on, or streets you can't even drive down. None of them will have any parking. Hotels do not have parking.
Drop the car somehwere like Heathrow. Or, depending where your last night in the West Country is, drop it ther and get the train back to London.

Driving in London:
https://www.youtube.com/watch?v=E0k92wHArFE&t=13s

it should be noticed that that video above is for outer suburban driving - in South London in this case, in and near Penge to South Norwood - which is dramatically easier than driving in zone 1 or 2, Central London.

Seconded what Nigel said. That is suburban driving, not central London. The roads are (relatively) empty, and there is on-street parking, and there are very few pedestrains.

Compare to this (Google streetview of the Strand in London, probably on a relative low-traffic time like Sunday): https://www.google.co.uk/maps/@51.5106315,-0.1210731,3a,75y,100.87h,86.41t/data=!3m6!1e1!3m4!1seVNMUYkJz8ma-w1lhQgM5w!2e0!7i13312!8i6656

" for one day before flying out from "

Does this mean in practice you will drive from X to London, drop the car, check-in to your hotel and then, the next day go to Heathrow? If so, your time in "London proper" is limited and it might be less of a faff to go to Heathrow, drop the car, use a Heathrow hotel and then just go into London by train for the rest of the day/evening before going back Heathrow for the night. Especially if you have big luggage or an early flight. An alternative could be to drop London entirely this time and head to a city where you can drop the car and then visit before going direct to Heathrow the next day by coach or taxi. For example Oxford or Windsor/Slough.

" for one day before flying out from "

Does this mean in practice you will drive from X to London, drop the car, check-in to your hotel and then, the next day go to Heathrow?

Most likely. I think we have given ourselves enough time to make it worthwhile, as we'll have most of two days--as long as we don't spend the entire time returning the car. I was not surprised to see overwhelming advice against taking the car into the city centre. Your idea of a drop-off in Oxford sounds like a good, viable back-up plan.

I'll rephrase the last question: does anyone know about returning a car other than at Heathrow? As a random example, I see that Budget has locations in Westminster and Nine Elms which would be much closer to our eventual destination than that long ride in from Heathrow, but would that really shorten up our total travel time?