I've always been under the impression that Elizabeth II genuinely served as a military mechanic. However, I recently realized that I have not personally read credible accounts of her acting in that role beyond showing off her responsibilities to her mother during a royal visit which, of course, could merely be part of an honorary role. I understand it's not uncommon for dignitaries or famous individuals in the military to serve in honorary roles?
I am aware that, at the very least, she slept in Buckingham Palace each night rather than with the other ATS members.
I answered a similar question a while ago here.
In short, yes she did join the ATS and did participate in training. However, from there her role was mostly symbolic.