"Never mind" in American English and British English

Solution 1:

I'd say there really isn't much difference in usage between American English and British English. In both dialects, it can be used to mean "it's alright", and dismissively to mean something like "shut up, I'm annoyed". A lot of it depends on tone of voice.

Solution 2:

Considering @Snubian's answer, I think the phrase could be used both ways in both regions.
I don't see a particular UK/US distinction; rather, the difference depends on the mood of the speaker, as @Jez suggested.

I do notice a difference (but again, not a regional one) between transitive and intransitive use:

Transitive = Disregard

Fred: I'm slightly worried about X
Bill: Never mind that - what about Y?

Intransitive = "Oh well, that's a shame, but let's not worry about it"

Fred: So I'm afraid we'll have to have pizza, not tacos
Bill: Never mind. I like tacos.