It sounds like these folks would have been members of Lonodn livery companies (Worshipful Company of...)
Records of these are often very handy as they cover how they became members (e.g. through apprenticeship, or through their father).
For example, here are all the Coopers listed as apprentices in Worshipful Company of Carpenters records from 1654-1694.
https://www.british-history.ac.uk/no-series/london-carpenters/1654-94/pp201-224Cooper, Edward, 21, *152, *190; Elizabeth, *81; Francis, 180; George, *105; Godhelpe, 161; James, *180; John, *5, *134, 152, *161, *198; Prudence, *78; Rebecca, 156; Richard, 23, 40, 59, 76, 83; Thomas, 20, 156; William, 9, 57, 69, 78, *79, 81, 86, 91, 127, 147, 190.
entries take the form of:
Edward Cooper the sonn of Edw: Cooper of Stratford upon Avon in the County of warwick malster hath putt himselfe appntice to Christofer Lockwood for seaven yeares from midsomer day instant.