Churchill was pretty discredited following Gallipoli, and had no following or political base in Canada per se. So although he could go to Canada in such a scenario i doubt he would gain the premiership, especially over Mackenzie-King.
In terms of Independence, i suppose it depends on the nature of the British revolution. Canada was already independent in domestic matters, and gained full control of foreign affairs in 1931. The Canadian Monarchy is constitutionally separate from the British one, even though the offices are held by the same person, and it is the structures of the monarchy and Parliament, that determined the powers that Britain still had over Canada at the time.
So if the British terminate their monarchy and replace it with the Peoples Republic of Alba, or whatever they are calling it, they in essence cut Canada and the other dominions loose constitutionally. I suppose in such a case Canada would merely accelerate to the present day situation in terms of our governmental structures being fully separated from Britain's, with the addition that our monarch, whoever he/she would be, is no longer monarch of Great Britain as well. Possibly the Monarch would also live in Canada full time but that is not guaranteed. I don't know what would happen with Canada vis a vis the other Dominions and the Empire. Almost anything is plausible.