London universities and Bristol are clearly very much more well-known than BPP. If you have one of those on your CV then legal employers will recognise it as a good university. BPP is more of a risk as only a few cohorts have graduated - I think the first '3 year degree' lot graduated last year so they will be on the LPC/BPTC now. No employer knows how good (or bad) the degree is yet.
If I was an 18 year old looking to study full-time I would definitely choose, say, UCL over it - assuming that that is a realistic possibility for you. The course is fine, if you're choosing between mid-tier universities charging £25k and BPP charging £12k then I can definitely see choosing BPP. Another factor to bear in mind is that BPP does not feel like a traditional university. There are no halls, and there are less social events / clubs. There will probably not be a big group going on a pub crawl in Fresher's week.
I'm a mature student doing the 2-year degree at BPP so happy to answer any questions about it, or you can search through my posts as I've answered many of the most obvious questions already.