I can help you a lot with Badminton, in the sense of advice.
I'd recommend starting by finding out if there are any clubs near by, whether or not they represent a Borough or County, and then joining them. You could join your school at the same time. Use the Internet for the club search; for the school one, ask your Sports teachers in school.
Just to note, you cannot just pick up a Badminton racquet and leap straight into a school or club team (forget a Borough or County team). It is a difficult sport to learn, and takes a lot of practice, and you will have to master many aspects of it; I'm assuming that you already know how to play reasonably well, which is why you're asking to join a team. If not, I would join a club anyway, but with aims to improve your play rather than start playing competitively. If you do play well, then still, join a club (from where you can be directed, if appropriate, to a Borough or County team); however, keep in mind that, having not played any competitive Badminton yet, you have no idea of how good players at County level really are. Every field of their play is what you would expect from someone who: 1) is considerably more talented that the average player, and 2) has put in as much hard work as anyone else, but had a lot more competitive experience than you.
The same would go, I presume, for Basketball.
School teams may be much easier to qualify for, because there may be much less people in your year trying to play Badminton than the number at a club. But still, expect a challenge (at most schools) if you've never played competitively before.