I can't find any record of any significant number of British adults "fleeing" the UK during WW2. In fact, the UK had a net gain of adult males emigrating to the UK from Ireland and the British empire to do wartime work. There were also ten of thousands of refugees or displaced people from continental Europe living (or interned) in the UK.

The two significant groups of British people who left the UK during WW2 were child evacuees and wartime brides. A plan to evacuate tens of thousands of children to Canada, Australia New Zealand and South Africa was cancelled after only around 2,500 evacuations due to the sinking of a ship carrying nearly a hundred children. Though by the end of the war, around 14,000 children had been evacuated privately to relatives or foster homes abroad, mostly to Canada and the USA.

War brides and their children also left the UK from 1942 onwards, though the vast majority left after the end of WW2. I don't have any figures to hand.

There was significant urban to rural migration within the UK, as well as evacuation of coastal towns on the south and east coast of England - around 3.75 million civilians were displaced at some point during the course of the war.

There is a possibility that people fled from Northern Ireland to the Republic of Ireland during the war, but those people would have been Irish citizens and likely identified as Irish.

Source: Wartime: Britain 1939-1945, Gardiner 2004