Although the brooch on the Royal Magazin site was a wedding gift to the Queen from the Overseas League, the idea of giving the then Princess Elizabeth the brooch was initially proposed as a 21st birthday present. In early May 1947 the UK media and those in Commonwealth countries carried reports that the League had opened a shilling fund to raise money for the brooch. I assume that, as the brooch had not been presented by the time the Queen's engagement was announced in July 1947, the League sent it as a wedding present instead.
I have never seen a photo of the Queen wearing this brooch, but I presume she would have worn it when she and Prince Philip visited the headquarters in 1948.