I've been up and down Blackheath Road any number of times, but hadn't spotted this ghost sign until I walked past it a few days ago. That's perhaps not so surprising: its location on the side of the building, with another house alongside and luxuriant plants in front, mean it's not exactly conspicuous. That suggests that these were the business premises: it's hard to imagine the wall as a paid-for advertising site.
Indeed, the 1911 census confirms that the house was occupied by the Taylor family. Head of household Edwin Taylor was an auctioneer, but his 21-year-old son Hugh was already an engineer. Presumably it was Hugh who went on to build an engineering business here with his own son, suggesting that the sign is from the 1930s or later.
The right-hand side of the sign is no longer legible, but enough remains to give a sense of what was being advertised. On the triangular gable, 'Taylor & Son. Civil [illegible] mechanical engineers' was painted in red-brown. Below, larger black letters spell out 'Contra... Plumb... Sanitar...' suggesting a plumbing business as well as the engineering one.