Learn English – Come in front or Come from front

Nice one is quite informal and has a couple of meanings. It's often followed by mate, (a friendly form of address between men) in BrE and AuE as in Nice one, mate. In AmE the term buddy could be used instead of mate.

When used in the positive it means Well done or Thanks (or both) here's a few typical examples.

You take your car to the garage to have the brakes fixed, when you pick it up the engineer says 'The brakes are done and we also adjusted your clutch', You reply 'Nice one'. This is the well done and thanks meaning.

You are in the pub and someone buys you a drink, when they give you the drink you say 'Nice one' This is the informal thanks meaning.

You are watching football and someone scores a good goal. You say 'Nice one'. This is the informal well done meaning.

All of the above are positive and likely to be what you heard in the movie but it's possible to use Nice one in a negative way. Here are a couple of examples.

You are in the pub and someone spills your drink. You say 'Nice one' but you mean That was stupid and careless of you.

You are telling a long joke to a friend and another friend cuts in and says the punchline long before the punchline should be said. You can say 'Nice one' to express your disgust at your friend spoiling the joke.

You can really only tell the difference between positive and negative Nice one by the context and more so by the tone of voice used by the speaker.

I wouldn't say it was specifically British English, Australians definitely use the positive version and as noted in the comments Americans use both the positive and negative meaning but might not use it for thanks.