Most efficient way to get from British Museum to LHR

Walk to Tottenham court road underground station and take the elizabeth line to LHR

My suggestion: Walk to Tottenham Court Road Station and take Elizabeth Line to the airport. This assumes you will go directly from the Museum. However, I have to ask, "How will you manage your luggage while at the Museum?" I am assuming that you will need to go to your luggage storage location prior to heading to the airport.

The most efficient way would be to take the Elizabeth line from Heathrow to the Tottenham Court Road station. Get off there and walk about 5-10 minutes to the museum.

EDIT: Oops, I misread your post. You're looking to go from the museum to LHR. :-) In that case, just reverse.

We did this exact thing on our departure day recently. We stashed our luggage at a shop right across from the Tottenham Court Rd tube station for something like 5 pounds per bag (found on an app called Bounce, I think?), went to the museum for a couple of hours, then took the train to Heathrow. A big advantage to this route is that you don't need to take steps or even escalators from the street to the train, there are lifts that will take you where you need to go.

If you are arriving on the Cal Sleeper you aren't allowed to use the Avanti First Class Lounge. You use the Sleeper Lounge on Platform 1, open from 0630 to 1030.

You also could take the Piccadilly Line Underground from Russell Square direct to Heathrow. A bit slower than the Elizabeth Line but that location might be more convenient. And it costs less. If you’re open to alternatives to the British Museum the British Library treasures gallery is very worthwhile, very close to Euston, and I believe it opens at 9:30 (later on Sunday). The Charles Dickens house museum is also not far away. You could probably see both of these in their entirety in the time it would take just to see some highlights at the BM. If you do go to the BM the line to get in tends to be shorter at the northern entrance on Montague Place than at the front on Great Russell Street, as many people don’t know there is a second entrance. Also you often can get in faster if you have no bags of any kind, even smaller ones you could take in, with you so don’t need to wait in line for that screening.

The Treasures Gallery will be closed from 18th August to 27th September, should that cover your dates.

As Carol mentioned above. You need to figure out where to store your luggage. That will be your departure point heading to Heathrow, not the British Museum.