I've had similar thoughts to varying extremes... on the one hand, occasional extended deployments (say 405 Sqn RCAF deploying to Australia with their Argus ASW aircraft for a year, or an RAAF Mirage squadron deploying to Comox for NORAD ops and suchlike)... to the most extreme, a basically unified Commonwealth defence policy...
...this would see the RAF, RAAF, RCAF and RNZAF (and, until some point in time, assuming everything is the same as in the RW timeline, the SAAF and RRAF) each having their own specialisations. Naturally, each would have some units for 'local use' for defending one's airspace, army co-operation and whatnot, but the rest would all be part of the specialisations, such as:
RCAF's specialties would be ASW and heavy interceptors (assuming the Arrow is built!), so you'd have RCAF units of Arguses and Arrows more or less permanently based in Canada, Australia, the UK, NZ, South Africa, Aden, Malta etc...
RAF's would be nuclear and heavy conventional strike, so squadrons of V-bombers based in the UK, Canada, Australia, Rhodesia/South Africa, etc...
I suppose the RAAF would share the ASW load with the RCAF (how does an Argus with Roo roundels strike you? I know it tickles me!), perhaps along with Canberra/F-111-level conventional strike... and RAAF and RNZAF could split the anti-shipping load... maybe with, say, an RNZN squadron based on RAN carrier(s)?
But playing more with your idea, I like it, and we could maybe see an RAAF fighter squadron based in Canada contributing to NORAD efforts - say, for supply-simplicity's sake, with CF-100s first, later Voodoos (Roo roundel F-101s... yum!), and based in the UK, an ASW/MP squadron operating Shackletons and later Nimrods?