Bristol is usually considered the next best maths course after the Cambridge/Oxford/Warwick/Imperial cluster. Doing more difficult mathematics earlier doesn't necessarily mean anything in terms of academia, since, especially in mathematics, it's critical to have an excellent foundation in the core areas of (pure) mathematics. How well they prepare you in these core topics of e.g. analysis, abstract algebra, and linear algebra is pretty crucial to becoming a good mathematician (more so than doing some more esoteric mathematics early).
If you are mainly interested in pure maths or have no particular inclination yet (which is perfectly reasonable) then I'd suggest Bristol. If you are fairly certain you want to go into mathematical physics or applicable mathematics (i.e. statistics, applied probability etc) then UCL is probably equally decent an option (possibly slightly better on the applicable maths front if you go for the maths/stats course there, since they have an entire separate stats department contributing to it). Durham is really probably running behind these two unless you really want a collegiate uni or want to do NatSci while being able to take proper pure maths modules, simply due to fewer possible options.
That said the three of them offer the usual standard options at the relevant level, broadly speaking, so you won't be unduly disadvantaged in a career as a mathematician going to any of them. Bristol has the widest range of options for the 4th year (assuming you're looking at the MMath/MSci versions of the courses) of the three, which correlates to its usual perception of being the next best after the "top tier" courses. Durham has the narrowest range of the three. However they all have the essential material available to potentially go into pretty much any area of maths for a PhD.