if you look at pictures closely (such as this one), you can see that nicholas is taller. alix's hair might reach more, but that doesnt really count. it was, however, during their engagement but growth probably would have stopped by then.
Note though that you can't see their feet. Plus, Alexandra's shoulders are higher than Nicholas's.
Alexandra's perceived height difference to Nicholas may indeed be down to her wearing slight heels and having elaborate hair, but in most photographs Alexandra is noticeably taller than Nicholas, even when she doesn't have a hat on, or a fancy up-do.
I don't know their exact heights, but I do think Nicholas was around 5"6 and Alexandra around 5"7 or 5"8. I read quite frequently that Nicholas didn't have the tall, strong figure of his male ancestors; he was short and slight, a real contrast to his bear-like father.
Rachel
xx