I was originally told by a regular in The Blythe Hill Tavern that the building in question had been a public bath's.
I perpetuated this myth until whilst parking up outside 24 Blythe Vale I commented to the chap trimming his hedge that he had a very interesting house. He informed me that this particular house was the market gardeners when "it was all fields around here". His knowledge seemed quite extensive, it transpired that his father had an estate agency many years ago - a Tudor looking building on Catford Bridge, and I asked him if he knew what the bookies in question used to be. He confirmed it was associated with the market garden and was in fact the despatch building from where the produce was transported, presumably on carts. I assume that there were ramps up to cart level within the building with the carts backed in below the large arches, this is obviously all guess work.