You are right about the difference between British and American English with regard to the words 'revise' and 'review'.
In British English, 'revise', in a school context, means going over the coursework that's been studied and learning it in preparation for an upcoming exam. It's used more for what students do alone, rather than for a class activity.
'Review', in British English, generally means looking at something in order to give an opinion of it, such as a book review, or in order to update or make changes to it, such as a salary review.
That said, I'm sure it would be clear what a teacher meant by 'Let's do a quick review of some grammar points'. Context is everything when it comes to language - and in this context it would be obvious that they meant 'looking at it again briefly'.