It wasn’t ‘unusual” for Canadian Army and British Army (and other Nationalities, Like the Polish Armoured Division, which was part of the First Canadian Army for some time) to be “lent out” to each other..........when the units were serving together, of course they all supported each other. And many major operations had a mixture of Units from different Armies (British, Canadian, US etc) supporting each other.
If you read the history of the Canadian’s in Italy, you will see that they were part of larger British and American Armies and served with and alongside of New Zealand, British, Indian, US and other troops.
During the entire war, Canadian soldiers, sailors and airmen were attached and part of other Allied Units all over the world.......and they to our units.
Two “Aircraft Carrier’s” (Nabob and Puncher) were Royal Navy ships.....but were “crewed” almost completely by Canadian’s (except for their “air wings”).
And of course Canadian airmen served in almost every RAF Squadron, around the world.
AND, you can’t forget about NEWFOUNDLAND......they were not part of Canada during the war (they were still a British Colony) and their men and women served in British Army units throughout the war (They raised many Artillery units that served in North Africa etc etc)