There are numerous coffee shops, cafes, snack bars and tea rooms in the Central London area where you will get a good hot drink.
A personal favourite of mine for a great cup of tea on the go, quickly and costing next to nothing, is the green taxi-drivers cabin on Hanover Square, Mayfair. These cabins provide hot meals and drinks for London cabbies, but will serve anyone a tea and a bacon sandwich from the serving hatch - lovely jubbly! The taxi cabins are dotted all over London, watch out for them, and try a cup of tea that won't cost a fortune.
Traditionally, the best places in London to get a good afternoon tea - sandwiches, cake and tea - have been the big hotels. If your budget stretches to it, try the Ritz, the Savoy or maybe the Cafe Royale - all are renowned for their afternoon teas, though you likely won't get much change from a twenty-pound note for your "high tea".