Hey there,
I've lived in bath for about 6 years now as an undergraduate and postgraduate. In my experience, the nightlife is far from dead but it looks a little different from other unis.
On-Campus (University of Bath)
On campus is The Plug and Tub, the university bar and nightclub. It is very popular with students, especially first years and masters students. They have regular events and lots of societies have socials there.
Bath
Bath itself is regularly busy with nightlife. There are fewer nightclubs than when I started my degree, but still some. I think the difference is you get a mix of students, locals, and tourists at these places, whereas at other unis you would have places that predominantly cater to students.
Bristol
Lots of students head to Bristol for nightlife (myself included). Most tend to go in second year onwards, as in first year most make use of the plug and tub on campus. There are lots of places people go to, and you tend to get a more student community than staying in Bath.
I hope that answer helps!