By the 50s and 60s, the ownership of telephones was increasing, so if you don't have an address, start with the 'phone books on Ancestry. Armed with that, most Local Studies Libraries/Archives will do a quick lookup for free, so try e-mailing them before tackling the British Library. Even if you do need to visit London, you'll probably find the local library less daunting than the BL, and they'll have the ERs on the shelves as well as other local records the BL won't have. But check that you're going to the right place, as local libraries are unlikely to have ERs for neighbouring boroughs.
But if you give us more detail (without giving the names of people still living) someone here may live close enough to help.