Princess Alice's choice seems unusual. And even more so since her family had a number of grand tiaras which would have been suitable. On the other hand, perhaps a crystal headpiece might have been the height of fashion.

This article on her wedding gown might give some insights into her choices. https://www.vogue.com/article/the-royal-bride-who-wore-pink-princess-alice-duchess-of-gloucester Perhaps the simplicity of the gown was meant to be a foil for the headpiece.

I haven't collected any cuttings of her bridal attire, but I do recollect many saying that she wore the string of pearls which her recenly deceased father had given her as a wedding present.

The pearls