Tabanus: Based on 10 Leicester Square, the <R.> can't be right ..1823: <Arthur Joseph> born Apr: 2, William & Elizabeth <Simons>, Leicester Square, Silk mercer (Church of England Births and Baptisms, St Anne Westminster Middlesex)
1841 census Leicester Sqre 10 & 11 (St Anne Soho Middlesex): Willm <Simons>, 53, Silk mercer, Elizth Simons, 45, Eliza 20, Maria 14, <Arthur J> 15, + a shopwoman, two milliners, etc.
1851 census <10 Leicester Sq>, Saint Anne, Westminster (Middlesex): Elizabeth <Simons>, Head, Widow, 57, Milliner employing 3 persons, <Arthur>, Son, 27, At home in Bas-----; Mary, Daughter, 29, Maria, Daughter, 24, and 3 servants/milliners assistants.
1853: Letters from <Arthur Simons, 10 Leicester Square>, in British Chess Review, Vol. I, pp. 251-254.
1854: <As no reply has been received from this man Simons; or, from what we can hear of him, is ever likely to be, ...> (Chess Player's Chronicle, Vol. 15, p. 89)
1854: <Arthur Joseph Simons>, Tobacconist in Murray Street, father's name William Simons (deceased) Silk Mercer, married 6 Dec 1854 in St John, Hoxton, Hackney to Catherine Mary Ann Wells (Church of England Marriages and Banns)
1859: <Deaths ... On the 25th Oct., at Port of Spain, Trinidad, Catherine Mary Anne, wife of Arthur Simons, and second daughter of the late Fredick. John Wells, of North Down, aged 27.> (Morning Advertiser, 2 December 1859, p. 8)
1866: <Deaths ... On the 26th April, at St. Vincent, West Indies, aged 36, Ann Frances, wife of Arthur Simons, and eldest daughter of the late Frederick John Wells, of Northdown, near Margate; alaso, on the same date, her infant son, aged 3 days.> (The Canterbury Journal and Farmers' Gazette, 16 June 1866, p. 4)
After this, having lost two wives (which were sisters) and one child in 6 years, he disappears. (There are many Arthur Simons's.) He may have stayed in the West Indies, left for US or elsewhere, or (more likely) returned to England. In the latter case, here are two candidates: