IIRC, Princess Elizabeth drove a jeep around during the Blitz, which was a somewhat shocking thing for a Royal to do (though, at the time, she was not thought to be in the short line for the throne).
And she was Heir Presumptive at the time, and given that the only way she could have been displaced would have been for her mother, already in her forties, to give birth to a son, it was fairly certain and accepted that she was to be the next Monarch.
But at the same time, the succession was secure - Princess Margaret was available (and still young enough that she couldn't get away with doing what she liked at all), with the Duke of Gloucester (who would have made a fine King if needed) next in line after her.
The fact is living in London in the Blitz was dangerous anyway - Buckingham Palace was bombed. The sensible thing to do would have been to send Elizabeth and Margaret to Canada. Having made the decision to stay, allowing Elizabeth to take some risks isn't that surprising.
She was the Heir Presumptive, in a very short line - she would be the next monarch unless her mother (then in her mid 40s) had a son.