They probably just rent studio space somewhere, so there's no real physical base (other possibly than a small office, and perhaps a basic production suite, somewhere).
^^^ 'London TV' seems to be nothing more than a rather poorly-designed website, showing video content hosted by Youtube and using widely available press releases.
'London Live' is the only station licensed by OfCom to provide local broadcasting for the capital. As Hopkirk points out, it's based at the Evening Standard's offices.