That's interesting about Omega using the Barr and Stroud sleeve valve engine, and also the oil-cooled Bradshaw. Around 1920-22 Omega were using (at least) JAP, Villiers and Blackburne engines, but this was clearly not enough because the model they announced for 1922 used a 3 hp two stroke that they built themselves. This bike was weird all over, with a widely-spayed duplex frame, unusual front fork, unit construction and modern styling. The advert below describes the new machine, but is illustrated with the very conventional 293cc JAP model which matches the bike in Graham's photo.
Leon