I should know better than to comment before checking around a bit, including in my own books. Sir Alan Lascelles' (Private Secretary to King George VI at the time) diaries of 1943 and 1944 mention the issue in some detail, and so does the BBC.
BBC NEWS | UK | Wales | Royal plans to beat nationalism There was considerable discussion about it in the press and amongst politicians coming up to Princess Elizabeth's 18th birthday. I can't pinpoint whether it was Welsh nationalism that precipitated consideration of the issue, but it seems it may have been, and thus Elizabeth was being considered as a handy political pawn.
Lascelles' view was that constitutionally the remote contingencies, of the King & Queen having a son or the Queen dying and the King remarrying and having a son, could not be rejected as negligible, and on 27 September 1943 he informed
The Times' editor, Robin Barrington-Ward, that, should there be a serious demand for making Elizabeth Princess of Wales, he would be safe in reminding his readers that she was, after all, only Heiress Presumptive, not Apparent, and to let them draw their own conclusions.
Lascelles commented on the issue further on 8 January 1944 in connection with correspondence received on the subject, and which he sent on to the King. He thought there would be a lot of talk about it in the following four months and expected it to be brought up in the House of Commons, the Commons having the right to approach the Sovereign on the subject, citing the example of the petition by them to Edward III in 1376 to make Richard of Bordeaux Prince of Wales. On 13.1.44, Their Majesties consented to Lascelles writing to the Lord Chancellor to draft a press announcement that the King did not propose to change Princess Elizabeth's style and title for the present. There had apparently been a "spate" of "press comment and general chatter" on the subject and Lascelles thought it a good idea to nip it in the bud.
As an aside, I find it interesting that at this point Lascelles noted, "It is so much easier to do this before the waters have started to rise than to bury your head in the sand, and trust that the flood won't incommode you, which is the usual technique of the Royal Family in the face of such threatened agitations".
On 27th January Cabinet was discussing the matter and came to the conclusion that if anyone was going to make an announcement on the subject if should be the Home Office, not Buckingham Palace. Lascelles did not care which one did it, noting "my motive throughout is to get somebody to make some decision before an undignified controversy erupts in Parliament and Press. But I suspect that some of the Cabinet rather favour her being made Princess of Wales - possibly the PM himself, whose pictorial imagination is no doubt fired by a vision of Princess Elizabeth looking charming on the steps of Carnarvon Castle, and blushingly acknowledging the loyal acclamations of a pan-Welsh Eisteddfod. That, of course, is all very nice; but I doubt if any of them have really thought out the full implications of such a step - particularly what I described in a letter to John Martin (Lord Chancellor) as the physiological contingencies".
On 2nd February Lascelles described events at a luncheon that day, and noted that Churchill had dissuaded the King from making any definite pronouncement to the effect he didn't propose to change Elizabeth's title, preferring to leave it to Brendan Bracken (Minister for Information) to "damp down" any Press publicity about it. Lascelles was disappointed, calling it a reversion to "ostrich" tactics, "which won't stop the Welsh going on being troublesome, nor private individuals from writing foolish letters". Making Princess Elizabeth Princess of Wales was very appealing to some politicians as "bait to catch Welsh votes".
It is noteworthy that that evening Lascelles joined the King for dinner with the PM, three Chiefs of Staff, General Eisenhower, Gen. Walter Bedell Smith, General Omar Bradley, and Field-Marshall Montgomery. There were, after all, a few other things going on in Europe at the time, particularly plans for something that would eventually take place on the 6th June!
9th February 1944 Cabinet discussed the announcement to be made about Elizabeth and it was agreed it would be made from BP, not No. 10, and on 10th February Cabinet "made no bones about the Princess Elizabeth announcement (that there be no change), and, having cabled it to the four Governors-General, I hope to get the wretched thing published here on Saturday", which it was.