For computer science Ds sat a maths entrance exam for Cambridge which is the CTMUA. The maths test is TMUA but Cambridge pay for it and the results are provided directly to them hence the C in front of TMUA. The entrance exam was taken at the beginning of November of year 13.
Ds did not have to sit any other entrance exam for any other uni. Unfortunately there are no direct links to provide for you as such. There are university guides but each one differs as to where they rank universities.
The process for Ds was, decided he wanted to do computer science, he had taken it for GCSE and A level and did other CS related stuff outside of school, loves it. Googled top universities for CS, made a list. Decided he didn't want to be more than a 3 hour drive from home so that ruled out London unis and Scottish unis.
Then you need to look at predicted grades for A levels and where your child can realistically apply. Durham (where Ds is now) their entry grade is AAA however, Ds was predicted A A A A. Therefore a very realistic chance of getting in. This is the important bit, there is no point applying to Warwick who ask A A A for computer science if you are predicted AAA.
Sadly the whole process of looking into universities etc comes from researching. It isn't just about the university though, or the course, but location too, city based or campus based? You need to visit to get a feel for the place. Look at the cost of accommodation. Price really varies. Also look in maintenance loan amounts and expected parental contribution which may also play a role in determining a university.
So the whole thing starts with research. Ds made a massive spreadsheet, complicated slightly more by both Cambridge and Durham being collegiate where you apply to a specific college there rather than just the university.