If you haven't been to London before, you might be surprised with how walkable the center is and how small the central neighborhoods are.
I agree that Bloomsbury (where you are staying I believe) is a bit of a culinary wasteland. I lived in Bloomsbury for three years. But it really doesn't matter. Soho is right next door as is Fitzrovia and Covent Garden - all areas with many more options.
For example, Great Queen St. in Covent Garden is 0.4 miles from the British museum. Soho square is 0.5 miles from the British Museum.
Hmmm... it's a little bit of a vacuum over there. Mostly hotels and not-great restaurants. You might have an ok higher end pub meal at the Marquis Cornwallis on Marchmont st. Ciao Bella does a good Italian on Lamb's Conduit st. For a gastropub, nearest one I can think of is the Lady Ottoline on Northington St. Kind of depends on exactly where you are staying in terms of how close these would be.
If you are actually close to the museum, think about searching restaurants in Fitzrovia (the next neighborhood over to the west.) British Museum / Russell Square area is not great for dining options in my opinion.