Hey I'm going into 4th year at Bath so let me know any questions!
In terms of how fast further maths is covered, before christmas you'll study 5 different modules at the same time. One of these is called Methods, which pretty much covers the further pure maths you need (calculus techniques and differential equation solving). There are two modules, algebra and analysis, which are there to introduce "proper" pure maths (haha) which means proofs and rigour about mathematical statements. So they treat theory of sets and functions, some basic number theory and group theory, limits and continuous functions, from a rigorous point of view. Modules differ depending exactly which course you're on but the other two modules may be probability, of which the content may be similar to a-level but again treated in a rigorous way, and discrete maths, which introduces algorithms and programming.
The main point of the first semester is to introduce a kind of new way of thinking about maths, so it may be going over familiar material in a new kind of way (ie proving all the statements made)