Alastair Arthur was born in 1914, the son of Prince Arthur of Connaught (grandson of Queen Victoria) and Princess Alexandra, duchess of Fife (granddaughter of King Edward VII). He was the first great-grandson of Queen Victoria in male line to be born in the United Kingdom. His birth registration (see a copy here) designates him as a "Prince". Several contemporary references (Kelly's Handbook, Whitaker's Peerage) style him as prince.
Sometime in late 1916, the duchess of Connaught asked the Earl Curzon to look into the style of her grandson, the infant Alastair. Curzon contacted the Lord Chancellor discreetly, avoiding Buckingham Palace (he wrote that "The King is indifferent [?] rather hostile, having always been rather jealous of the Connaughts"). The Lord Chancellor replied on Jan 11, 1917 that "it would be in accordance with usage that the son of Prince Arthur of Connaught should be styled 'Prince' and 'Highness'" but cautioned that no step should be taken without consulting the king.
Presumably such consultation took place, and on March 23, 1917, the king's private secretary indicated to the home Office that "His Majesty's wish is that he should be styled "His Highness Prince Alastair of Connaught". A warrant was prepared to carry out HM's wishes, but before it could be issued the king, considering the changes he was about to make in these matters, asked that the question stand over for the present. (see files LCO 2/7299 and HO 45/18980).
By the terms of the letters patent of 1917, Alastair was not allowed the style of Prince or the style of Highness, since none had been granted to him formally. Burke's Peerage ("The Princes of Great Britain", 1963 edition, pp xxvii-xxxii) considers this to be an injustice. It is not clear that this result was initially intented: Lloyd George's instructions of Aug 29 on the drafting of the letters patent, when coming to the 3d generation from the sovereign, give as example: " Thus in the event of further children being born to Prince and Princess Arthur of Connaught, they would he Lords or Lady...Windsor" (emphasis added), suggesting that Alastair Arthur was not expected to be styled Lord Alastair Arthur Windsor. (see HO 144/22945). But the article of the Times of June 20, 1917 announcing the changes that would be implemented by the Letters Patent of 1917 states that "should he succeed his grandfather and father, he will be Duke of Connaught, but not his Highness nor his Serene Highness".