Well, Britain was allied to Austria at the time. So it's not unthinkable that one of George II's younger son(s)? The duke of Cumberland was George's favorite, to the point that he attempted to even split the British and Hannoverian inheritances. So, perhaps Georg Wilhelm (George II and Karoline's second son, survives, so pushing Cumberland still further away from the British throne), and as a way of making nice with Russia, Britain offers Cumberland. Anton Ulrich was a disaster as a general, yet held the title of generalissimo of the Russian army (something that didn't go over so well with the Russians due to his incompetence), so perhaps Cumberland might make a decent substitute.
OTOH, Anna Leopoldovna was making eyes at the Saxon ambassador (can't remember the guy's name) before her wedding to Anton. Her aunt, Anna Ivanovna requested the Saxon Elector to recall the ambassador, and the minute Ivan VI became emperor, Anna Leopoldovna recalled him. He was halfway across Poland to take up his post in St. Petersburg when the coup of Elizabeth Petrovna happened. So he just quietly went back to Dresden.
Frederick the Great was also mentioned as a candidate for her hand at one point - but I think that that was when she was still only Princess of Mecklenburg, not a grand duchess of Russia