(Not a teacher)
I would say the most you could say is whether their accent is 'more like American' or 'more like British'. Non-native speakers who have a perfectly identical British or American accent do exist, and for them it would be appropriate to say they have a British/American accent. Such people normally go through elocution lessons, however; such an accent doesnt come by learning the language.
Considering the amount of variety and accents within Britain and within the US, I dislike referring to them as British/American accents, however, learners seem to like this distinction.
There are, of course, large communities where English is widely spoken. South Africa, Austrailia, New Zealand, India, to name a few. I wouldn't express these as being 'more American' or 'more British'. They are simply where they are from.
Similarly with my fiancée, who is a native Brazilian Portuguese speaker. I would call her accent Brazilian English; it doesn't sound like anything other than her Brazilian accent speaking English words. I would do this for all speakers of English, as there is no doubt that your original accent exists independently from the language you know.