The idea that a real Roman bath - as used by Vespasian's legions, even if not by Caesar's - was still in use in Strand Lane, just a few steps down off the Strand, probably started as an advertising ploy in the 1830s. Before that, from at least 1776 onwards, the establishment was advertised as just a cold bath, sometimes attached to Number 33 Surrey Street, sometimes an address in its own right. But it's a powerfully seductive idea and it isn't surprising that it has lasted as well as it has - helped especially by an early puff from Charles Dickens in David Copperfield, but before that by a prominent place in Vol. II of Charles Knight's London (1842).
Here's how J.W. Archer depicted the bath for Knight, with an engraving from a watercolour he made in 1841:
Here's how it was made to look more authentically 'Roman' in the 1890s:
The notice to the right of the gentleman in the bathing costume still survives, even though the rest of the decoration was stripped away in the 1920s, because by then it didn't seem Roman enough:
And here is how the 1920s 'restorers' - the Rev. William Pennington-Bickford (Rector of St Clement Danes), Edward Foord and Fortunino Matania - dreamed that the bath would look when they had finished with it:
The Bickford-Foord-Matania plan, built on the conviction that the bath belonged to a suburban villa that they were also going to excavate and restore, was crazy and soon foundered for lack of funds. But we need to do something to rescue the bath from its present state of dowdy semi-neglect. Even if it never was Roman, we'll lose something if we don't cherish all the dreams of Rome that have attached themselves to it. And what it really is - the last Central London survivor if the eighteenth- and nineteenth-century fashion for cold bathing - is exciting enough in itself.