I've never commented on this sub before though I'm a big fan. I don't have any flair and am not an expert or anything, BUT I happen to know the answer to this question and can document it. Please forgive me mods if I break any rules or need to change anything.

In a nutshell, the British did in fact make plans for the royal family to be evacuated to Canada in the event of an invasion, though there was some pushback from the Americans on making Canada the seat of the government-in-exile, as the Americans were uncomfortable with the prospect of a monarchy operating in North America.

From Andrew Roberts' The Storm of War,

Although Britain's gold reserves were transferred to Canada, and plans were made for the royal family, the Cabinet and ultimately whatever was left of the Royal Navy to follow them, it was not even certain that the British Establishment would be universally welcomed by the North Americans. Ever loyal Canada was sound, of course, but on 27 May 1940 Churchill's private secretary, John 'Jock' Colville, noted in his diary that the British Ambassador to Washington, Lord Lothian, had telegraphed that afternoon to say that President Roosevelt had told him that 'provided the Navy remains intact, we could carry on the war from Canada; but he makes the curious suggestion that the seat of Government should be Bermuda and not Ottawa, as the American republics would dislike the idea of monarchy functioning on the American Continent!'

Source:

Roberts, Andrew. The Storm of War (2011)

Edit: Addendum: If anybody is interested, the British also discussed the possibility of evacuating large numbers of troops to Canada as well.

It has been revealed that soon after Dunkirk Anthony Eden and the new Chief of the Imperial General Staff, Sir John Dill, convened a secret meeting in a hotel room in York which was attended by the senior officers of formations based in the north of England. The War Secretary asked whether the troops under their command 'could be counted on to continue to fight in all circumstances.' Brigadier Charles Hudson VC recalled that 'There was an almost audible gasp all around the table. To us it seemed incredible, almost an impertinence, that such a question should be asked of us.' Eden explained that in the circumstances the Government were envisaging, 'it would be definitely unwise to throw in, in a futile effort to save a hopeless situation, badly armed men against an enemy firmly lodged in England.' They would have fought on the beaches, it seems, but not so far north as York.

The subsidiary question that Eden and Dill put to the officers was 'Whether our troops would, if called on, embark at a northern port, say Liverpool, while it was still in our hands, in order to be withdrawn to, say, Canada? Without such a nucleus of trained troops from the Home Country, the Prime Minister's declared policy of carrying on the fight overseas would be infinitely more difficult.' Hudson related that it soon became very apparent that the officers were all of much the same opinion. While the proportion who would respond to the call among Regular officers would be high, and of Regular NCOs and men who were unmarried nearly as high, 'No one dared, however, to estimate any exact proportion amongst those officers and men who had only come forward for the war; a smaller proportion of unmarried men might respond but the very great majority of these would insist on either fighting it out in England, as they would want to do, or on taking their chances whatever the consequences might be.' The upper reaches of the British Army were therefore of the view that the majority of its troops would refuse to embark for Canada to continue the struggle from abroad, just as many French had not embarked for Britain for the same reason earlier that month. It was all the more vital, therefore, to prevent the Germans from landing in the first place.

As an extension, what would be the likely fate of the royal families if the Nazis captured them?

In addition to /u/apeman2500's well-written response, one should remember in relation to this question that during WW2, Newfoundland and Labrador were not part of Canada, but were directly governed by the UK.

To ask a follow up question:

Did the British monarchy consider evacuating to Australia (or any other commonwealth countries)? Was Canada the only consideration, and if so, for what reasons?