You are right to be concerned.
• For their Management BSc, Warwick want "45 Credits at Level 3, including Distinction in 33 Level 3 credits and Merit in 12 Level 3 Credits".
• For their Business Management BSc, Birmingham make no mention of a Access to HE Diploma and being an acceptable entry qualification and say
here that "Access to HE qualifications will be considered by some programmes. However, this is usually on a case-by-case basis – check our course finder for more information." That doesn't sound promising.
• For their Management BSc, Nottingham suggest that they might accept an Access to HE Diploma but point you towards
this page which says that for an AAA course (which this is) they require 36 distinctions, 9 merits, and 0 passes.
Unfortunately, the fact that you say, "I feel I make a good student" won't be of much help. It's demonstrable academic performance, as measured by grades and exams, that they're looking for. You might need to set your sights a little lower, uni-wise.