Brady's was a chain owned by Brady Luke Ltd. wine and spirit merchants. I learnt the hard way, in the days before mobile phones, that when you agreed to meet someone at Brady's you had to specify which one.
I drank regularly in Brady's in Brixton Road (aka The Russell Hotel) and occasionally drank in Brady's Atlantic Road (aka The Railway Hotel). There were also Brady's in Shepherds Bush, Maida Vale, and the Euston Road. There may have been others but I think 5 is enough to make it a chain.
It was, or rather to my surprise, still is a family owned company and you can see their website
here. This explains that they sold out to Courage's in the 1980s and are now running a fish and chip shop in Wandsworth.