Getting to the British Museum by public transport
The Tube (Underground):
The nearest tube stations are at "Tottenham Court Road" (on both the Central Line and Northern Line), "Goodge Street" (also on the Northern Line) and "Russell Square" (on the Piccadilly Line). However, of the three, only "Tottenham Court Road" has step-free access. If arriving at "Tottenham Court Road" on the Northern Line, it's a simple case of rolling off the underground train on your own. However, you will need to use a ramp to leave the train if arriving at "Tottenham Court Road" station on the Central Line (and you must notify the staff of this - before you board the train at your DEPARTURE station - so they can call ahead and arrange for the ramp to be available/arrange for any assistance that you require).
So arriving at Tottenham Court Road, and on the Northern Line, is the easiest option by Tube.
A number of different buses run past, or near, the British Museum. And all buses in central London are wheelchair accessible - and are equipped with a retractable ramp (automatically controlled by the bus driver). Bus travel is also free for wheelchair users (your companion has to pay the correct fare though). I recommend that you use the official TfL Journey Planner to find the best route for yourself.
By far the easiest way to arrive at the British Museum in a wheelchair, using public transport, is in one of London's famous black cabs (taxis). It's also the most expensive way to reach the British Museum (but is so worth it in my opinion). All London taxis are equipped with a retractable ramp.