You don't need to watch BBC for that. Not a lot of tv shows, even child programs are dubbed on Dutch tv. So you get accustomed to it early on. Combined with the fact the languages are closely related means the Dutch usually have a reasonable grasp of English before it's being taught in school.
I learned english at school in France, and we're notoriously bad at teaching foreign languages. The approach is way to academic and mainly based on reading. That's why our accents are often atrocious. I was good at written tests, but what allowed me to actually get fluent (as in being able to think in english and convert my thoughts to speech in real time) was watching tv series in english with subtitles in english (no translation involved.)
