This wobble in the chainwheel is inherent in the basic design, and all Plunger Clutches have an element of axial movement at the chainwheel periphery. It's all down to the poor lateral support for the chainwheel, together with the need for a running clearance between the clutch basket and centre. Add a bit of wear here and there, it can appear terminal but still continue to perform with no problems. They all do it, even with new parts.
As a matter of urgency if you folks in a similar position have any doubt at all I'd recommend removing the pressure plate and clutch plates, to give access to the centre nut, which should be tight. This nut bears on a tab washer, 67 3253. The clutch centre should be fractionally proud of the ends of the mainshaft splines, to allow the nut to clamp the clutch centre against the thrust washer behind the chainwheel. The tab washer inner fingers bend towards the gearbox and locate in the small distance between the face of the clutch centre and the end faces of the shaft splines. Removing the nut and primary chain will then allow the whole clutch to be removed from the gearbox mainshaft for inspection. Then follow RD's suggestion, the thrust washer is supported on a split collar, without it the thrust washer is free to run against the oilseal rivets. This collar was actually missing from my bike when I first bought it! Look for witness marks where the chain has touched the chaincase interior. A slack chain and a loose clutch can machine away the primary cover screw bosses..to quote Musky.
Plenty on the Forum about the Plunger Clutch and watch the order of the plates..... the thicker plain plate goes into the basket first.
BSA's other 6 Spring Clutch with the weaker pressed steel centre addresses the issue of wobble by supporting the chainwheel with its friction segments against the rearmost plain plate, which is a robust and altogether heavier example, clamped securely onto the clutch centre adaptor when all is assembled and the centre nut tightened down. This is perhaps the only redeeming feature of this cost cutting and inferior design.
Swarfy