Mathmatical ability goes hand-in-hand with computing science; I'm not doubting that. However, the range of jobs that a degree in CompSci (even a conversion course) is varied, and not all of them are going to require explicit mathmatical knowledge or ability. My brother has a humanities degree, did a CompSci converson course and, even though he 'only' has an A-grade at A-level (and hasn't done a lick of maths since), he is currently employed. Maths certainly helps, but if needing a knowledge beyond A-level was an explicit requirement of studying CompSci and getting a job, then Universities would be outright asking for it; as it stands, people are being accepted with A-level maths (or even lower), passing the course, and finding employment. Granted, they likely aren't going to work for highly technical companies writing programs that require high-level calculations, but there is still ample opportunity elsewhere.
I'd say the significance of maths and programming is they both employ a similar mindset. You'll have the data (whatever it is) in one state, need to get it into another, and hence will have to abstract and visualize the steps required to get from point A to point B. Having a mind geared towards maths (as evidenced by good grades at A-level/degree) will help with programming, and if you have always struggled with maths, you might also find coding difficult. Ultimately though, more mathematical knowledge is always better, but inherent mathmatical skill is just as important - even if your job doesn't require you to program in-depth calculations, you will be relying on a similar mindset (really though, even a C at A-level indicates some aptitude, so I wouldn't be strictly worried).
Honestly, she should go into programming because she enjoys it; sods to whether or not she has the right background or not. Far more important than maths is doing it for the love of it, and not because she wants a job - any job - because *then* she may really stuggle. Also, don't discount the importance of being willing to work your arse off. People always forget that.