Founded at Hyde Park Corner by Richard Tattersall (1724–1795) it stayed in the Tattersall family until about WW2. The business had to move from 'the Corner' due to the lease running out and the land being allocated for the development of Grosvenor Crescent. So in 1865 the business moved to newly built premises on Knightsbridge Green. The picture showing the premises in 1938 is remarkably similar to the picture on Wikipedia dated 1865.
See the 1876 home of Edmund Tattersall (1816–1898), who joined the business in 1851, at the magnificent RBJC Local Studies.
British History On-line has more pictures and says: "Between 1865 and 1939 Tattersalls' yard was the scene of regular weekly, and in the season twice weekly, horse sales, events as much social as commercial where the bluest-blooded of aristocrats rubbed shoulders with the shabbiest of sporting 'characters'."
At the outbreak of WW2 the business moved to Newmarket where it is still based. The premises on the Green were bomb-damaged in WW2 and demolished in 1955, but the original arch, and the drinking fountain at the centre (see picture) known as 'the Fox' were both salvaged and re-erected at Newmarket. We haven't found the arch but the Fox seems to be a popular prop for wedding photos.