All 3 schools use their own aptitude tests and assessment process. That's why BA have the final stage of selection.
Oxford will only tell you that your scores are suitable for a self sponsored course if you've met the standard required for entry to the APP course - itself a very high standard that makes you eligible for Oxford's skills net guarantee where they will refund the entire cost of the course if you fail and there are no resit fees. They certainly don't give that to anyone turning up with a cheque book as it could cost them tens of thousands. The pass rate for the APP is around 50%. BA FPP is a competition so will only take the top x % above this.