At 17 you'd be either in year 12 or 13 (depending what month you were born), which is in sixth form/college (english college - not university).
Unlike american high schools which you graduate at 18, here secondary school technically finishes at 16 after GCSEs, then people either do an apprenticeship or go on to further education at sixth form (which is different from, but often attached to, a secondary school) or college doing A-levels or BTECs (though I don't know if those are the same or different to your BTEC). These are usually 2 year courses, and these grades are what universities consider when you apply.
If your BTEC course is the same as ones here, you could potentially continue that course at a college/sixth form here if they have that course, but I think it's more likely that you'd have to re-do/re-start year 12 (as you might start different courses) and then do year 13. Other people also sometimes have to re-take year 12 for a variety of reasons, so you won't necessarily be alone in that boat.
And yes, when you do your final exams at the end of year 13 many people your age will be starting university, but it's also worth noting that some people take a gap year after finishing A-levels and only actually start university the year after that (you can apply for deferred/delayed entry), so if after year 13 you go straight into university in September, some of the people you are studying with might be your age anyway.
Hope that helps