The Royal Family had plans to evacuate Britain in 1940, and purchased Hatley Castle in Victoria, British Columbia to use as the King's Royal Palace for a Government-in-exile. However, it was decided that the hit to morale for the British public would be too great, so the Family decided to stay behind.^1

A rather unreliable looking website states that they would have been evacuated to the Midlands and then protected by a civilian army in the case of a German landing, followed by an evacuation to Canada; however I can't access the source listed on the page so I'd take it with a grain of salt: http://www.birminghammail.co.uk/news/local-news/secret-war-plans-for-royal-family-222718

It seems most sources agree that if the UK had been invaded, the Royals would have eventually made their way to Canada and ruled from there for the remainder of the war.

Very odd. I was just reading about this today, though it's by no means my area of expertise. What prompted me to read about it was the story of Malcolm Campbell, who led the military police contingent of the Coats Mission, which was the evacuation plan for the royal family. Campbell (and this is why I was reading about him) broke the world speed record on land and water multiple times in the 1920s and 30s. I wonder if it was deliberate that the fastest man in the world was part of the mission to evacuate the British royals...