As far as I know, BSc and BA usually don't matter as much when it comes to architecture. I'm not as sure about Liverpool, but Bath is highly ranked in the UK (if something like prestige is a factor for you) and is a much more technical course, focusing a lot on the practicalities of architecture. A thing that also sets it apart from other architecture courses is how you get a placement year, which means you get experience before you finish your course - this would help inform your studies better and also possibly improve your chances at getting a placement as a graduate.
In terms of nightlife, I heard that it's not as good as places like Liverpool, but it's definitely still there (albeit smaller as it's a smaller city). I doubt you'd get bored due to this, and honestly, with how gruelling the majority of architecture courses are, I doubt you'd get much opportunity to party a lot. Though if you do plan on going out a lot, Liverpool may be more to your taste. Bath is great for sports though, so if that's something you'd consider alongside your studies as well as partying, I'd recommend that.
If you go to Liverpool, it doesn't automatically rule you out from getting a placement - so many things come into play when applying for placements, like location, etc, so you can definitely still get placements if you're a competitive enough candidate. Employers may be more familiar with Bath, though, so you might want to consider that potentially.
Overall, the course content and vibe should speak to you the most. Personally, I would pick Bath over Liverpool, but there's so much more to consider for your specific situation, so take this with a grain of salt. Good luck and don't sweat it too much! I'm sure you'll enjoy either of them