23 is a low LNAT score - not enough for a competitive Uni like UCL, and most LNAT Unis will also read/score your LNAT essay. This probably explains the rejections.
Notts in particular is a top Uni for Law. What was it you didnt like or is this actually just about vanity and you see yourself as a 'better' student than those at Notts? Be honest about this.
You need to work out if you actually want to study Law at all, and work through the over-focus on UCL. I'd suggest that you Firm/Insure with the two offers you have, wait for Results Day, and on the basis of the grades you achieve, make a decision about going with any Law place you may then have, or taking a gap year and reapplying (with more work on LNAT skills) - or A level resits if needed.