How about the Horseshoe (just off Snowsfields) - has a back room with table and very comfortable chairs for old bones.
And the jukebox can't really be heard from in there.
The food's not bad and the beer seems ok too (very vague recollections of this)
Just don't go on the first Sunday of the month (not to play bridge anyway) as there is an English Folk jamming seesion (which is fantastic - lots of accordions and the odd helican and serpant too!)
Try "The Lord Clyde" just off Marshalsea St, about 114 m from Borough tube station, maybe the 3rd or 4th street up on the right. I stopped in yesterday on Alan's recommendation and was pleased with menu, beers, human space, decor, old fashionedness. (But what do I know? I'm Merkan.) There's a room off on the right with space for bridge players, and the entrance is sufficiently user friendly to allow access for even the most elegant and elaborate of zimmerframes.