See Castle Street East 1 to 15 &c in London Addresses dataset
No 2 Castle St East Land Tax 1791-1800 Thomas Cook / SunFire 1794 Cook gent
A common name but I suspect, because it is so near the artist colony of Newman St, this may have been the engraver Thomas Cook ODNB 1744/5?-1818. Very little about his life in biographical sources. Thomas Cook, London subscr 6 copies of Bourrit's Glaciers of Savoy 1776 / Thomas Cook engraver, Stoke nr Norwich subscr 1779 to History of Thetford / TC Bartholomew Close took apprs 13.8.1783 for 7 yrs John Tennant Lovelace & Samuel Springsguth / Thomas Cook engraver Bartholomew Close 1786 prosecuted at Guildhall by association of copperplate engravers for practising that art without having served an apprenticeship in it (Cook won) 21.1.1786 London Chronicle / TC St Bartholomew the Great took appr 1.10.1785 for 7 yrs Edward Shirt / Holdens 1802 Shacklewell / Mr Cook engraver Upper Titchfield St, Fitzroy Sq ad in 17.12.1804 Morning Post / Holdens 1805 Gt Suffolk St, Charing Cross / his obituary Examiner 25.1.1818 son Henry promising engraver / Thomas Cook engraver of Cumming Place, Pentonville will dated 2.12.1815 all to wife Eliz who proved 6.7.1837 when he was late of Duke St, Westminster Rd, Surrey