"The teacher does not speak English."
“不會說英語” Should likely be acceptable. It's not like the teacher is refusing to speak English.
会 is "can" in a capability sense. 我会说英语。 "I can speak English" or "I have the ability to speak English."
“老师不会说英文” Should also be acceptable, because they are unable to speak English
In my opinion, 英文 (written English) does not match very well with 说. Instead we use 讲 (jiǎng, tell), 讲英文. 讲英语 is okay too.
A better way of writing that in Chinese would be "这位老师不会说英语." While Chinese is all about reducing complex concepts into the shortest phrases possible, as the sentence currently stands, it's awkward because we don't know which or how many teacher(s) (since both singular and plural form of teacher is 老师) and whether or not s/he is choosing not to speak English or does not know how to speak it.