The degree certainly doesn't help: when you're doing an initial sort of 100+ CVs and you have dozens of applicants studying actuarial science, stats, or numerical degrees, it's easy to see how a musician could end up in the rejected bucket from the initial screening. At that point, there is no feedback to give, beyond "we looked at your application for a good fifteen seconds and rejected it".
What they're looking for is certainty about your maths skills and (to be honest) even an A at A-level doesn't provide comfort in that regard these days. I'm not sure how much a few more CT passes would help, and it could well be a distraction.
In terms of internships, it may well be worth phoning up HR departments before your application lands with them to talk through the basics of it - that may be enough to get you out of the immediately rejected pile. Also, don't be afraid to consider other work in financial services - *any* office experience is a strong plus. And experience of the sharp end of client services is no bad thing either!
In terms of your overall strategy, I'm not sure what I'd do. If all goes well, you may end up with a job offer through internships.
If not, you either apply for graduate entry and cross your fingers, or undertake further study.
Graduate entry has the problem, again, that many firms specify a numerical degree, and again you will get binned at the initial screening stage.
Further study has the problem that it takes a lot longer. However, if you really want to do it, one suggestion would be to take a year out and find a role somewhere in financial services. Take the OU's Diploma in maths (I think with A-level you will only need to do the level 2 courses, which you can cover in a year). Go back to uni, have a crack at graduate entry. If you don't get in at this point, that (I would have thought) will be sufficient to get you on a postgraduate actuarial course.