It's interesting, because, historically, since 1914, anyway, Canadians have been in the forefront of efforts to assert our national control over our armed forces, even when they were tightly integrated into the UK's command and control structure.
My recollection, hazy at best and I'm too lazy (bad cold) to go do the research, is that there were deep divisions in Canada's delegation in London over Op Husky (the invasion of Sicily) with Gen Andrew McNaughton (grandfather of LGen (Ret'd) Andrew Leslie) opposing Canadian participation,* while others, including Vincent Massey, then High Commissioner, favoured a Canadian role.** The decision to go was, finally, made in Ottawa, by Prime Minister Mackenzie King, over McNaughton's objections and the issue of Canadian control became a factor in McNaughton's eventual return to Canada.
_____
* In part, at least, because he wanted to preserve a single, large Canadian Army for the (inevitable) invasion of France.
** In part, again, because Canadian public opinion, a factor of great significance to Mackenzie-King, was greatly in favour of "getting into the fight," somehow. Some of McNaughton's subordinates also favoured dividing the Canadian Army and joining in Husky,
as a very junior partner, because they felt a need to gain practical battlefield experience. I think Massey included their comments in his final report to King.
Edit: format