Le Cafe du Jardin in Wellington Street is quite quaint and the food is decent. You are a few minutes walk to the theatres the OP mentioned and they have very reasonably priced pre-theatre dinner options.
Banquette at the Savoy may be an option but I have not been yet and I haven't heard much about it either.
Le Deuxieme in Longacre is popular for pre/post theatre; and there is always Wagamama in Tavistock Street at Southampton.