What are the typical a-level grades needed to become a stockbroker, and where can I find relevant experience and internships
I don't think they typically focus on grades.
So long you have a suitable level 4 qualification in stockbrokering and you can sell/have superb people skills, I don't think the employers will care about your grades or A Levels.
Having said that, I would strongly recommend you complete your A Levels prior to going into stockbrokering as doing A Levels outside of college is a complete pain.
In terms of internships, you're better off looking at stockbrokering firms in the London. There are some in other cities, but London is the go to. There are also smaller stockbrokering firms is large-ish towns in half decent neighbourhoods, but you would have to really look for them. You would ideally want a medium to large sized firm for internships, as the smaller firms tend not to offer internships.
I don't know of any relevant work experience that you can do since it's mostly regulated selling, and you can't sell financial securities legally without getting the qualification. In fact, the qualification alone should help you a lot.