I took taninamdar's advice to look up the sources from the Wikipedia article in order to answer my own question.
One reference in the article refers to four other sources: "Bradford, p. 45; Lacey, p. 148; Marr, p. 100; Pimlott, p. 75". Each of these authors has published more than one book on Queen Elizabeth II. Googling I found the following books:
- Sarah Bradford, Elizabeth: A Biography of Her Majesty the Queen;
- Sarah Bradford, Elizabeth: Queen Elizabeth II : her life in our times;
- Robert Lacey, Royal: Her Majesty Queen Elizabeth II;
- Robert Lacey, The Queen: A Life in Brief;
- Andrew Marr, The Real Elizabeth: An Intimate Portrait of Queen Elizabeth II;
- Andrew Marr, The diamond Queen : Elizabeth II and her people;
- Ben Pimlott, The Queen: A Biography of Elizabeth II;
- Ben Pimlott, Queen: Elizabeth II and the Monarchy.
At the library I was able to find Robert Lacey p. 137 (not p. 148, so different from Wikipedia), Royal: Her Majesty Queen Elizabeth II, and Ben Pimlott, p. 75, The Queen: A Biography of Elizabeth II, so I refer to these in what follows.
One of the Wikipedia sources is a scanned image of the London Gazette 9 March 1945 (Wikipedia says 6 March 1945), listing under the title "AUXILIARY TERRITORIAL SERVICE", "H.R.H. The Princess Elizabeth (230873) Col. Gren. G'ds. is granted the hon. rank of 2nd Sub., 24th Feb. 1945." "Col. Gren. G'ds." refers to her previously having been appointed Colonel of the Grenadier Guards (in 1942 according to Wikipedia).
Lacey & Pilmott confirm that "2nd Sub." means 2nd Subaltern. The "hon." refers to the rank being honorary. Lacey describes her as embarking on the Vehicle Maintenance Course, learning how to service and maintain army vehicles at Aldershot and continues,
"...[She] was eager to get to know her coursemates. But while they
slept in huts at the all-female base, Elizabeth was chauffeured back
to dine and sleep at Windsor. Every lunchtime she was 'whisked away'
by the officers to lunch in their mess, and at lectures she was placed
in the middle of the front row, with a protective sergeant on either
side."
Lacey refers to the diary of Corporal Eileen Heron, when he adds,
"By the end of the three-week course on 16 April, Elizabeth had
managed to escape from her over-protective mentors and take tea with
the other girls. 'These cups of tea are getting a nice chatty
institution,' noted Corporal Heron, 'She talks much more now she is
used to us and is not a bit shy ... [She] says she will feel quite
lost next week, especially as she does not know yet what is going to
happen to her as a result of the course.'"
"She told Eileen Heron that
she was hoping to join ATS headquarters later that summer as a junior
officer, where she would have worked in an office with young women on
transport organisation."
"Less than a month after her course ended came VE day - 8 May 1945.
There was ATS work aplenty in the months of demobilisation that
followed, but George VI wanted his daughter back home on royal duties.
He did not see her future as working in an office, even a military
office, alongside other women, and Princess Elizabeth bowed to his
wish."
Lacey p. 138 continues by saying Princess Elizabeth wore her ATS uniform on VE day when she, and her sister with a group of Guards officers slipped away from Buckingham Palace to mingle with the celebrating crowds, and walked down Birdcage Walk, up Whitehall and round Piccadilly to the Ritz Hotel.
Pilmott says,
"The rank was an honorary one, but the training in driving and
vehicle-maintenance she underwent at No1 Mechanical Transport Training
Centre at Aldershot, was genuine."
"'The Princess is to be
treated in exactly the same way as any other officer learning at the
driving training centre,' maintained the official report at the
outset. [5.3.45, The Times.] To back this up, the Queen requested that
photographers should not be given any facilities. [1.3.45, RA GVI PS
7423]"
"If it was not quite true, as a 1957 assessment put it, that 'the rule
of seclusion was maintained and she did not mix with her fellows on
the course,' [unpublished obituary, 1957, Manchester Guardian] the
extent of mucking in, on equal terms, was limited. She kept to the
routine of the ATS mess, took her share of duties, and acquired the
basics of driving, car mechanics and maintenance. But she returned to
Windsor [Castle] every night to sleep."
The policy on not granting facilities to photographers seems to have been short lived. Pimlott describes Princess Elizabeth as becoming
"an unwitting mannequin for the uniform of the service - pictures of
her with a spanner, at the wheel of a lorry, leaning on a bonnet, or
peering purposefully and fetchingly under one, appeared in the
newspapers and magazines of every Allied nation."
These look like the photos in the link mentioned by Michael Borgwardt and AllInOne http://mashable.com/2015/04/22/queen-elizabeth-army/#i0MgZnvSakqz which according to the article are from March 1945 (which would have been during her training). King George VI and Queen Elizabeth, and her sister Princess Margaret also appear in some of these photographs. Pathe newsreel footage https://www.youtube.com/watch?v=R2grMaRttws shows Princess Elizabeth competently driving a Red Cross lorry or ambulance during her training, and shows her in overalls working on a Red Cross lorry during a visit to the training centre by her parents King George VI and Queen Elizabeth, and her sister Princess Margaret.
Pilmott thinks it is impossible to disentangle her private motives from the public effort:
"Since the enrollment of a royal princess could not be kept secret,
her participation in the ATS inevitably became part of the
morale-boosting display of the Monarchy."
Pilmott says a few days before the final end to the war she was promoted to Junior Commander, which is confirmed by the Wikipedia source showing a scanned image of the London Gazette 3 Aug 1945 (Wikipedia says 31 July 1945) "Hon 2nd Sub H.R.H. The Princess Elizabeth (230873), Col., Gren. G'ds., is granted the hon. rank of J. Comd., 26th July 1945."
