Hi. I'm currently studying for an economics degree and I know many people who are applying for banking internships (as well as applying for a few myself). Your GCSE/ A-level/Undergrad subjects will not stop you from getting a job in banking. Whether that be investment/commercial banking or a public sector role, it isn't just the economists and mathematicians who are applying. However, you might not be able to apply for some more technical roles like quantitative analysis.
I know people who have secured jobs and internships coming from non-mathematical degrees like history and psychology. It is also not uncommon for people studying the natural sciences to decide on a career in finance. As long as you can show that you are fit for the role, the company will take you. Further to that, most companies will have their own assessments and interview processes rather than relying on your CV alone. If you can do well in those, then you've got a good chance of landing a role.
Some final words of advice. Even if you don't do A-level maths it is still a useful skill to have. Don't let your mental arithmetic get rusty and maybe consider some more numerical papers during your degree. Some of the application tests will be very numerical for certain roles. Finally, if you're feeling up to it, you should look to apply for spring weeks early in your first year of uni. You should also consider looking for internships or other work experience before you graduate, which will give you a step up over other applicants.
The long and short of it is, your A-levels aren't stopping you from going into a career in banking. There isn't a single combination of subjects that employers are looking for, they just want to see how well you fit into a role and what skills you have. Do some of your own research into the specific roles that you are interested in, keep an eye on their application windows, and I hope you enjoy your first year at uni!