For the English side, I've only ever heard this followed by “with”:
This message is chiming with voters frustrated by the government’s record on corruption.
Their personal accounts chimed with many women and prompted them to share similar experiences.
I did not like her previous films, but for some reason this one chimed with me.
These (made-up) examples all sound natural and unobjectionable to me. However, various dictionaries define “chime with” as “agree with, harmonise with”, and a message/account/film can't agree with a person/group of people, so perhaps “strike a chord with” or “resonate with” would be preferable in all three cases.
But “chime with” would definitely work (for me) in cases where the object of the sentence isn’t a person or group of people:
His comments chimed with the attitudes of many religious conservatives.
Baffling as they may seem to modern readers, these ideas chimed with the sensibilities of the time.
Hugh Bonneville’s sentence (without the “with”) does sound strange, though not so strange that it would necessarily have jumped out at me if you hadn’t pointed it out. The voiceover translation cuts in straight after he says “chimes” – I can’t hear how he continues his sentence (and am no good at lip-reading), so it is possible that he goes on to say “with viewers” or something.
I wouldn’t have suspected any BE/AE differences here, but you never know.
For the German side, I think “einen Nerv treffen” is exactly right.