The Guild/Livery Company relating to Tailors is the Merchant Taylors, although many people were members of guilds which did not relate to their actual occupation (often because their membership had been gained through patrimony, or because they had purchased membership in one of the high-status guilds - in fact, the Merchant Taylors is one of the Great Twelve.)
http://www.history.ac.uk/gh/merchmem.htm - Guildhall Library holds records for this company.
By the time you mention, membership in a guild was not required (this was only really ever a requirement, as I understand it, for the Square Mile anyway: somebody might correct me on this, though). In about the 1830s the rules were changed so that you did not have to be a member of a guild to be a freeman of London.
The three ways that somebody joined a guild were through apprenticeship, patrimony - so if his father was also a member of a guild, even if it wasn't the Merchant Taylors, he would have been granted membership automatically from that company. (There are some notable exceptions: the Watermen and Lightermen have never practiced patrimony) - and redemption, which is basically purchase.
