Hi there,
There are many universities with a good reputation that doesn't require the applicant to have completed an LNAT. Coventry University is one of them. The entry requirements are BBB. I didn't complete the LNAT nor did I study any extra alevels or complete an EPQ etc and yet still received an offer, because my personal statement was strong. I would also recommend looking at Aston University, UoB and University of Warwick, and University of Law. (However for UoB and University of Warwick you may have to double check whether they wanted a LNAT to be completed by the applicant.)
I hope this helps. (I'm also from the west midlands so all the universities I stated are within the West midlands, which I'm familiar with. I apologise if the choices are limited.)
Best wishes,
Narusha
Coventry University Student Ambassador