Hello Neil,
All my reference sources agree that the BSA Cadet was only ever available in .177 calibre.
It appears that this was also true of the later Cadet Major although an experimental example was apparently produced in .22. The cylinder size and swept air volume proved to be insufficient to give any useful performance in the larger calibre with this example so the option was not commercially pursued. It was not until 1959 in fact that BSA produced a break-barrel air rifle (Meteor) in .22 calibre.
Hope this helps a bit whilst not providing an explanation for your unusual calibre rifle. Perhaps a .22 barrel from a different model has been adapted to fit the Cadet.
Regard
Brian