Just about every subject can be studied at several universities, and many subjects (e.g. mathematics, English) are available at near-enough all of them.
You need to start your homework early, almost certainly in the year before that in which you will seek admission (i.e. if you are at a school or college, in your last but one year). Begin to get hold of the information you will need, such as university prospectuses and brochures. Go to some of the general open days that many universities organise; this will help you to get a feeling for what different universities are like. Similarly, you might find that some universities visit your school or college and it is helpful to go to these talks if you can.
You should look for a course that, in its detailed style and structure and content, that suits you. Check whether you are likely to be able to meet its entry requirements.The next criterion is the location, travel expenses, living costs, student culture, city lifestyle, whether or not it is campus-based, and so forth. All these are important matters, and only you as an individual can decide on them; but they are secondary to finding a course that you like. If you like the course, there is every possibility that you will do well (and enjoy yourself in general). If you do not like the course, you might find it very boring.
Because of the wide range of courses involving statistics, it can be quite difficult to identify them all. It is important to read university prospectuses carefully and to use any national enquiry services that are available. If you are at a school or college, you should be able to find the information in the careers library. Many public libraries and other information centres also carry the material.
Your starting point is likely to be UCAS. UCAS is the organisation that handles all applications to all universities and similar institutions (apart from the Open University) in the UK. Its "Big Guide" covers all courses in all subjects at all of them. It is indeed a big guide and yes, you have a lot of work to do! You can buy the "Big Guide" on the UCAS bookstore web site if you wish or, if you are at school or college, you will almost certainly be able to consult it through them. In case you should need it, here is the UCAS postal address: UCAS, Rosehill, New Barn Lane, Cheltenham GL52 3LZ.
You can search for courses on-line, by subjects or by universities or by geographical regions, on the UCAS web site (click on the "course search" button). Your search will give you a list of courses, each with its own link to further web pages. You will find that you need a lot of additional detailed information. You can get some of this from the UCAS site itself, especially from the growing number of entry profiles that give course details. You may also find that your school or college subscribes to some other national database services carrying this kind of information.
You can get prospectuses by post from the university or by visiting your local Careers section of a library. Information is usually also directly available on the university's web site. In most cases you can guess a university's web address: it is usually http://www.name of university.ac.uk but unfortunately there are some confusing exceptions. Information is usually available about the university as a whole and about the courses offered by the faculties and departments within it. In many cases, departments have their own information handbooks; try looking for an individual department's web site, or writing to the Admissions Tutor of the department in which you are interested and asking for a copy of the course brochure.
And do not forget the Open University. This is unlikely to be of immediate relevance to school-leavers, but might well be very important to prospective mature students who may need to study part-time by distance learning. The Open University does not provide statistics courses going up to full degree level, but it does have excellent introductory and intermediate material in statistics. It also provides a great deal of mathematics material, including full degrees in mathematics; this material, even if not taken up to full degree level, might be very useful in providing the mathematics background likely to be necessary in supporting the study of statistics.
Finally, there are professional examinations in statistics offered by the Royal Statistical Society. Again, these are unlikely to be of immediate relevance to school-leavers but might be important to mature students. Also, they are used by some organisations as part of on-the-job training. The Society does not provide courses leading to these examinations, but distance-learning material is available from the University of Southampton.