We drove there on a Sunday and parked in a residential square a few minutes walk from the museum. It was residents parking only on other days but ok on Sunday and plenty of spaces. We had a look first on Streetview and zoomed in on the parking signs (which we also double checked after parking).
There is parking on exhibition road (probably all residents only during week though), and I'd have thought some will be disabled spaces but I'm not sure. Failing that there's loads of parking in Hyde park just at the top of the hill. Frighteningly expensive of course but convenient and quite a nice walk back down to NHM (about 10 mins or so depending on speed and how close to the gates you find a space).
The last time we went we stayed at the Holiday Inn Kensington Forum, which does have some onsite parking for blue badge vehicles, but since then DH has had to get a more powerful (i.e. bigger) outdoor scooter and thus a van rather than a car As their parking is underground, I'm not sure what options that's going to leave us with
This thread from MoneySavingExpert has some interesting ideas and this maybe?