I've just finished my work.

In the following conversations, which one is correct?

A:
Lucy: I've just finished my work.
Mary: OK. Let's go shopping.

B:
Lucy: I just finished my work.
Mary: OK. Let's go shopping.

Both are correct. A is more natural in BrE. I believe B is more likely in AmE.

Piscean said:

Both are correct. A is more natural in BrE. I believe B is more likely in AmE.

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What's their difference in meaning or emphasis? In this thread #10, you said that “I just remembered” and “I’ve just remembered” present the situation from two different viewpoints. What are the viewpoints?

In BrE, the present perfect directs more focus on the present completedness of the situation. The past simple focuses on the fact this situation happened in past (however recent) time.

Lucy: I just finished. Work is done for the day.
Mary: Great! Let's go shopping.
Lucy: I want to go eat first.
Mary: OK. Let's go to Shomars.

@Maybo
I don't think there's a difference in emphasis in your sentences. I think the difference has to do with style.