The Blitz occurred when then Princess Elizabeth was only 13 and 14 years old - I think there's a lot around on the web that has taken the fact she was a driver (and the book you've quoted does convince me she trained on ambulances) and have extrapolated from that as to what her role was.
By 1945 when she was 18 and had joined the ATS, there wasn't that much call for military ambulances within Britain anymore. I wonder if that was why they might have been used in the training schools.
Could be...the K2s were reportedly very difficult to drive.