I do CompSci at Bath; the course is very good, and the uni's good too.
I would say do the AS in Further Maths, if possible. Almost all of your competition at top unis will have done Further Maths A-Level, so do the AS to at least reduce the gap. You can request and view university admissions data online on sites like whatdotheyknow to get some data to help you.
Here's some UCL admissions data for CompSci this year (Just an interesting fact: 200 of their offer holders were international and only 75 UK, which will be similar for most London unis). Less than half of their offer holders did FM A-Level but I'm presuming that most of their offer holders didn't do A-Levels at all, being international students. In statistics for
Bath, around 60% of students do FM. Take a look at similar stats for unis you're going to.
Just to note, if you want to get into UCL's CompSci program, you'll need to do a lot of work. I was predicted and achieved A*s in CompSci, Maths and Physics A-Levels, with an A* in EPQ and As in FM and Econ AS-levels, but didn't get an offer from UCL. Like you've suggested, do some large scale projects. For inspiration, look at what you'll learn during the courses at some of the unis you want to go to and do projects in those languages. Do a lot of related reading, lots of unis have lists of books to read. Volunteering, internships, insight days etc if you can get them are very useful.
Good luck, let me know if you have any more questions.