We really enjoyed the tea at the Royal theater. Tasty and beautiful room and not as expensive or tough to book as some others. Feels very traditional — not jazzed up like some of the themed ones.
If you want a pub, the one across from the British museum is historic and has really good food. We went for a late lunch and basically no wait. I think it’s called the museum tavern or something.
Not sure where you are staying in south Kensington but I thought this Italian place was really good. We ate at another great Italian place near the palace but I can’t think of the name.
https://cacciaris.co.uk/menu-south-kensington/
We also ate at a great hipster style place in notting hill —great brunch if you happen to be there.
https://www.cafebeam.co.uk/
Also there is a chain called Gail’s that we often stopped at for a quick sandwich. And of course Pret which is a London export to DC.