Children were evacuated to live with volunteer homestay families in the countryside. From what I was taught at school, it was a case of suitcase, gas mask and name tag, then get on the train with all the other kids. This is how The Lion, the Witch and the Wardrobe (and many other books/films) starts.

For those that stayed behind there was a post-war pride that they stuck it through and didn't give in, but it must have been unimaginable even with the bomb shelters and London Underground. Again, this is what I've been taught from my elders so not first-hand experience.

EDIT: I should add that I realise it was not just Britain for whom things were unimaginable.

EDIT 2: Another factor that must have had a big effect on Londoners was that the king and queen stayed at Buckingham Palace (which was hit a few times) during the war.