I'm an American speaker, but I'm not aware of any difference in American and British usage here. (See these Google ngrams for "glasses" vs. "eyeglasses": American, British.)
Generally, "eyeglasses" is uncommon (according to the above ngram links, and my personal experience) but will be understood perfectly. I would expect to see "eyeglasses" used when you need to differentiate between an ambiguous use of "glasses" (which can also mean "drinking cups, made of glass"):
"Have you seen my glasses?"
"Yes, I just filled them up with water." (This person assumes "glasses" here means "cups")
"No, I mean my eyeglasses!"
In fact, there is an old joke:
My grandmother is 90 years old, and she doesn't need glasses! She drinks right from the bottle.
The first sentence suggests "glasses" means "eyeglasses" (since eyesight degrades with age) and the second sentence reveals that "glasses" actually means "drinking glasses" (since she drinks from the bottle instead of pouring her drink into a glass).