They are both good so you probably cannot go wrong with either choice.
I am a lawyer and my sons are both doing the LPC with BPP currently and they know someone very well who is doing the UoL LPC currently. as the boys live at home and I work from home I probably have a bit more contact than when they were away at university.
BPP is fully open book eg you can take anything into the exam which is on paper, your own notes, books, anything (but nothing electronic other than the computer on which you are working). I do not know as much about UoL but it is probably open book too. With BPP you can choose if you do the exams proctored/via web cam or in person in an exam hall but in both cases you may take any materials in. For my sons who type very fast and find hand writing exams more difficult that tipped the balance in their choice of online exams over in the exam hall,although either is an option with BPP. I believe this academic year UoL has exams in the exam hall whereas BPP obtained permission from the SRA to keep the choice of online exams for those who want it for this academic year.
BPP provide a sample paper and examiner's report of the year before and also I think some video reports on certain sample assessments.
More of the bigger firms use BPP although UoL is an excellent LPC course too.
BPP is wholly open book even with MCQ including today's professional conduct exam.
For BPP online exams students may not print anything in the exam not even the exam paper and pdfs supplied and nor can you take electronics like a mobile in nor google search terms of course, but my sons have not found that a problem and found the online exams fine. We were worried as our internet here at home cuts out once a day briefly that might mess up the exams but it has never been a problem and they could have chosen in person exams had they preferred.
BPP holds the core subject exams in January (business law, property, criminal and civil lit, public law) and I believe UoL hold them about now in March. I prefer the fact BPP gets them over with a bit earlier on so by about mid Jan you have done exams worth about 63% of the year's marks, and then does the skills topics (and then the electives which are the balance of the marks as the electives do not feed into the overall mark but you do have be assessed as competent with them) but it is not a huge difference.
I am not involved enough to know if the exams are harder. I thought I heard UoL might be slightly different or trickier but that just anecdotal from my BPP twins who are not doing the UoL course. If I had to guess I would say those with high A level grades and/or sponsored by a law firm (whether at UoL or BPP) are probably both as likely to pass whichever they chose.
The pass rates by institution are not published so no one knows.
If you do not do the LPC with LLM option then BPP's course finished when the exams do in mid June (or earlier if you do the accelerated version) whereas I believe UoL finishes a bit later, but I might be wrong.