The difference between the American and the British version is not about "course", it is about "on" and "up". How does one get "on" a course? A person can get "on" a bus and can get "on" the fast track, but not "on" a class, or course. Would this work in BrE - I've enlisted up to a four year term in the Marines?
Perhaps this use of "on" is rather unusual but in the UK, one "gets on" a course by going to the website or the college and "signing up" for it. If your application is successful then you are officially "on the course".
We don't use it with "class" but then we don't really use that in the same way as it's used in AmE. Where you say "He's taking Italian classes", we say "He's having Italian lessons".
We don't say "enlisted up", but we do say that we "sign up for/to/as" something.
I've signed up for singing lessons this winter.
I've signed up to volunteer with clearing rubbish off the streets on Sunday morning.
I've signed up as a volunteer at a charity shop.
With the Marines, I think we say "I've enlisted on a four-year term in the Marines" but I'm not knowledgeable about military terminology. It might be "for", not "on". Or something like "I've enlisted in the Marines for a four-year tour".