I had to look up "Amazon Fire Stick", sounded like an exotic weapon in "Xena, Warrior Princess" (which surely would have made a thoughtful gift for your mom). And in so doing, I came across the Amazon UK site, which states this:
"Fire TV Stick connects your HDTV to a world of online entertainment. Whether you're a Prime member or not, enjoy a huge selection of movies and TV episodes, voice search that actually works, and exclusive features such as ASAP. Fire TV Stick is an easy way to enjoy Netflix, Amazon Video, BBC iPlayer, low-cost movie rentals, live and on-demand sports, music, photos, games and more."
(Bolding of the relevant part by me.)
Anyway, it seems to indicate you in fact don't need a Prime subscription for non-Amazon services, just whatever subscriptions/apps are relevant for what you want to use.