Yes, there was indeed a British single-cylinder motorcycle that featured a dummy second exhaust. The motorcycle in question is the BSA Bantam, specifically the earlier models. The BSA Bantam was a lightweight motorcycle produced by the Birmingham Small Arms Company (BSA) from 1948 to 1971.
The BSA Bantam's design was inspired by the German DKW RT 125, which BSA obtained as part of war reparations after World War II. The DKW RT 125 had a two-stroke single-cylinder engine with a distinctive twin-pipe exhaust system. To mimic the appearance of the DKW RT 125, BSA equipped their Bantam motorcycles with a dummy or "fishtail" second exhaust pipe on the right side. This dummy exhaust had no functional purpose and was purely for aesthetic reasons.
The dummy second exhaust on the BSA Bantam became a recognizable feature of the motorcycle, even though it served no practical function in terms of the bike's performance. Over the years, the Bantam underwent various updates and revisions, and later models did away with the dummy exhaust.
It's worth noting that the BSA Bantam had multiple engine displacements ranging from 123cc to 174cc, and different models were produced for various purposes such as civilian commuting, military use, and off-road riding.