Just here to echo some of what has already been said. I'm an A-level Computer Science teacher at a large Hampshire sixth form. In our local area there are very few 11-18 state schools. We have approximately 150 new students enrolled each year. Since the availability and quality of CS GCSE in schools is so variable, it wouldn't be fair for us to have any expectation of doing it as a prerequisite. We only require maths GCSE 6.
I would say that about 3 to 7 students per class (out of 25) have NOT done the GCSE. Since there is a lot of overlap between the GCSE and A-level specifications, students having succeeded at GCSE undoubtedly feel more comfortable to start with. Especially with programming, fresh students are often playing catch-up for at least half a year. However, after that, there's no reason why they can't be in line with the rest of the cohort (or, indeed, overtake them).
I would say that, at this time of year, after one half term, many of my non GCSE students are working at a C grade, sometimes far ahead students who have completed the GCSE qualification. By the end of the course, we have definitely seen non GCSE students achieving A's and A*'s at A level. We do also have some non GCSE students at this stage who are really struggling. Our analysis shows that performance in GCSE Maths by far has the strongest correlation with final grade (those with high maths GCSE grades are much more likely to get high CS A-level grades, though that's certainly not always the case).
Due to similar issues with availability and quality of CS A-level, universities offering theoretical CS degrees cannot require the A-level. Instead, they require Maths A-level, and recap a lot of the A-Level content (of course, in much more depth).
As many other people have already said, the thing that makes the difference for those non GCSE students, apart from Maths ability, is a strong work ethic and an ongoing commitment to understanding and learning the content. Getting a head start with the programming language studied in the sixth form you choose over the summer before year 12 certainly wouldn't hurt. You should be able to ask them for resources for learning that particular language.
The fact that the sixth form you spoke to is concerned, worries me. I wonder whether or not they would offer a supportive course that caters for all ability levels.