I can't get which one, class teacher or form teacher or homeroom teacher, in the UK and the US, is what we call "professeur principal" in French secondary schools, meaning one of the teachers of a class of 25 to 30 students who not only teach them their own subject, but also are in charge of their pupils as far as their overall school life and academic performance are concerned, checking out that they are doing well or helping them overcome the difficulties they might be going through.
Well for me this is how it goes.
Before and after school, in the Uk, in most schools, you have form where you take a register and do different stuff.
That is what I would call a form tutor.
Whereas someone who teaches lessons is a class teacher
In the USA this is called a homeroom and the person who is in charge is the home room teacher