The name of the company appears in professional publications, including Chemist & Druggist, on several occasions between 1870 and 1932, with most mentions dating from the last decade of the nineteenth century.
During that period sales must have been good enough, and Bates & Co, who operated from 1, Regent Square, London, opened a branch closeby at 55, Sidmouth Street.
I haven't been able to establish when Bates & Co was founded nor what happened to it after 1932. The houses on Regent Square weren't built until 1829 and the first inhabitants of number 1 were Isaac Seabrook, builder of St. Pancras Church, who lived there between 1834 and 1835 and 1838 and 1844, and Charles Templer Depree, solicitor, between 1848 and 1854. So Bates & Co was registered originally at another address and only moved to Regent Square at some point between 1854 and 1870, as the Chemists' Annual List from that particular year gives that address.
Bates & Co used this wall twice to advertise its remedies. Unfortunately the upper left corner, which is more exposed to the sun, has completely faded, leaving only a few letters on the right.
The most recent sign is on the left, the one in the background on the right:
For ...s [Wounds?] & Sores
... [Of?] All Kinds
... 80 Years
Citrate of Magnesia
Invented in 1844
The Original Safest
Location: Regent Square / Picture taken on: 01/04/2008