As a speaker of American English, "Have a nice day" is fine among colleagues but is a bit formal; I'd probably only use it with a stranger or with someone I only know a little. Among people I know well (close colleagues or friends), I'd be more likely to say, "have a good day" or even "have a good one."

As a native American English speaker I agree with Daren here. I would never say 'have a nice day' to someone I know and in business it's a very trite thing to say as well. To friends or colleagues I'd say 'have a good one', 'see you next time', 'take care'. Businesses that want to deliver a more personal experience would say the same or similar.

I don't know about the US or the UK, but I tend to say "Have A Nice Day" to people that I am not well acquainted with in order seem friendly. I can also use it sarcastically in way that basically indicates that I am annoyed or irritated and wish to exit a conversation.

In the United States nowadays, I think "have a good day" is more common than "have a nice day." However, you do hear both in real life.

I agree with what everyone else says. I will tell you a common use of this phrase. I hear and use this phrase A LOT in the elevator! I live in NYC in a large apartment building. I hear -- and say -- 'have a good day" or "have a good one" (more and more common) when I am in the elevator with other people, both when people get off the elevator before me, or I am getting off the elevator before them.