The Royal Exchange was destroyed by fire in 1838. Lloyd's is, perhaps, the oldest collective establishment in the City. It was first under the management of a single individual, who started it as a room where the underwriters and insurers of ships' cargoes could meet for refreshment and conversation. The Coffee-house was originally in Lombard-street, at the corner of Abchurch lane; subsequently in Pope's head alley, where it was called "New Lloyd's Coffee-house;" but on February 14th, 1774, it was removed to the north-west corner of the Royal Exchange, where it remained until the destruction of that building by fire.
A lot more early detail is at the London Taverns site.
A listing of historical London public houses, Taverns, Inns, Beer Houses and Hotels in St Bartholomew at Exchange parish, City of London.
Residents at this address.
1822/Bennet, White and Bennet, Lloyds Coffee House, Royal Exchange/../../Pigots Directory
1833-34/John Bolton, Lloyds Coffee House, Royal Exchange/../../Pigots Directory
1836/John Bolton, Lloyds Coffee House, Royal Exchange/../../Pigots Directory