Can you please indicate your predicted or achieved sixth form grades and your GCSE results, as advice about choice of university depends on your academic profile.
The University of Law used to be the College of Law, which provided the old style exams for those wishing to be solicitors. It is now owned by a Russian businessman and is a for-profit business, whereas many of the more traditional universities are educational charities. UoL is not research-led. It serves a purpose by providing PGDLs, and professional qualification courses. I would not recommend it as a place to obtain a law degree.
The University of London is nowadays a somewhat curious mixed bag, because its biggest colleges have more or less gone their own ways (UCL, LSE, KCL, QMUH etc*), but LLB (Lond) is a qualification worth having.
Would you not wish to have the university experience of learning with others, socialising with others, and pursuing sports and hobbies? Studying alone could, as you note, be a bit lonely.
If you plan to have a job while at university, have you looked at Birkbeck?
Looking ahead, it is SUPER HARD to obtain a place in barristers' chambers and it is also very hard to obtain a career as a legal academic at a good university. You should try to obtain your first degree from the most academically competitive and rigorous university you can obtain an offer from, and then consider an LLM or BCL and, if taking the academic path, a Doctorate.
If you are intent on study in or near London, and you are academically competitive, I would suggest UCL, KCL, and QMUH. I should disclose an interest and say that I teach part time at UCL, but at present I only teach LLM candidates. I hear somewhat gloomy news from a legal academic friend at LSE about the student experience. RHUL is less competitive, but a friend who teaches law there says it is a pleasant place to study. I also do some occasional teaching at Reading, and think it's an underrated university with a pleasant vibe.
*Imperial College has left the University of London, but doesn't offer law.