What do I need to do as a potential mature student in the UK? Please bear with me in this question despite its length. I'm only putting so much detail in it because I want to be thorough, as this is an important subject to me. 

I'll explain my situation first as to provide some context for my dilemma - I'm 19 years old, soon to be 20 in a few months, and I decided last year that I wasn't ready for university for a number of reasons. The most prominent of these were a fear of falling into debt, not being able to support myself while there and also not being sure of what career path I wanted to pursue. In light of that, I decided to apply for an apprenticeship at a fairly prestigious but small software design company in the field of Business Intelligence. I got the job (and am on the way to completing the level 4 diploma in "Information Technology, Programming and Networking) and will be employed until 21, when my contract ends. Over that period of time I'll be able to save up a fairly substantial amount of money (for me), 

I'm convinced now that I want to apply to a few of the top universities in the UK for Computer Science once my contract is over, but my A-level results aren't quite up to scratch for these. For reference, I have an A, B and a C in Computer Science/Psychology/Music Technology, respectively. What I needed was three A's. The universities I have in mind are Birmingham, Bath, the Imperial College of London, and Oxford and Cambridge - just on the off-chance I somehow get in or attain the necessary qualifications. The exact course that I've looked at is a five-year long MSci in Computer Science with a year of industry work/study abroad.

As a mature student though, I understand that I'll be subject to different entry requirements. One of these is an access to higher education course, but there aren't any within 50 miles of me that I can do a relevant course in via distance learning or even just evening classes, which means I'd need to do such a course after I finish my work contract in November 2016, further extending the amount of time before I can begin a degree. 

If anyone has any experience in applying to a highly ranked university - especially in the field of computer science - as a mature student, is there any advise that you can give me? 

As an extra note, I have emailed the universities in question to ask them for their advice and what would be expected of me when it comes to applying as a mature student. I haven't heard back yet, but I only sent an email last week.
I went to the UK to start my undergraduate when I was 27.  I was also not able to access the normal 'go-to-class' higher education courses.  My path wasn't computer science, but I think the general rules and requirements are roughly the same for each course.

In a mature student, UK universities are typically looking for enthusiasm, evidence that they are interested in the subject they are applying for, and that they are capable of doing university work.  Ideally what they want to see is that you have done some systematic course of study, typically in the last three years.  You have this (A-Levels and the diploma).  They also value experience - which you are getting by being in work in the area you want to pursue a degree in.  

Your A-Level results aren't bad at all, but as you note fall short of what they usually want.   With your work experience and additional formal certification, it is possible that the universities will look more favourably on your application.  Remember that even if you were to go into university straight from A-Level, grade requirements aren't hard and fast rules.  Which leads me to the next bit that is important - the personal statement.  This is the point at which you can say that you are more than capable of doing the course.  Don't dwell on your A-Level results, they can already see them.  Instead you sell yourself.  Really- this can make an application very attractive!  It's not all about grades - especially when you are a mature student.

If you did want to pursue another formal course, have you looked at the Open University options?  I did one of their courses (one 10 credit course) as I had been nearly 10 years out of education, and quite frankly my A-Level results were not...great.  It needed a lot of commitment to study and work full time, but overall it was a good experience, and I felt the course was delivered well.  If you can find something there that is relevant, I can highly recommend them.