London in May
Non-touristy, well you are in London one of the biggest tourist destinations in the world. These spots will have plenty of both locals and tourists. Perhaps this may help as a starter:
Café in the Crypt: near Trafalgar Sq., George's Inn in Southwark and Porters English Pub in Covent Garden, I think. Interesting venues with good food and drink and good fun, moderate prices for London. Lovely time to visit, best wishes!