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Hello there. You’ve certainly posted a tricky question!
Let’s first say that both expressions are correct. It’s perfectly alright to say ‘Are you finished at the flat?’ instead of ‘Have you finished at the flat?’ ‘Are you finished?’ uses the past participle ‘finished’ as though it were an adjective. So in a similar vein, we can say:
Have you finished at work? or Are you finished at work?
Both questions are grammatically correct and acceptable in any situation. Some native speakers would tell you that they see no difference between the two question forms. But others might say that they perceive a slight nuance of difference between them. It really depends on the tone of voice the question is delivered in and the attitude of the listener.
Let’s discuss this in more detail then. Some people consider ‘Have you finished?’ to be more polite than ‘Are you finished?’ In the examples I’m about to give you, either question form could be used but perhaps if you replace the question with ‘Are you finished?’ in the following situations, a bit of impatience is implied. See what you think:
Mum to kid eating — Have you finished?
Teacher to student — Have you finished?
Some people would argue that using the present perfect tense and forming the question ‘Have you finished?’ seems to politely give the possibility of more time to finish, whereas ‘Are you finished?’ lets the person know they are nearly out of time and no more is available.
Well, personally, I think tone of voice is the all important aspect here. You can express impatience or politeness just from the manner in which you say either of these questions:
Have you finished? Are you finished? (that was my impatient voice)
Have you finished? Are you finished? (that was my polite voice)
Well I hope you could hear the differences there and that you are successful sounding polite or impatient depending on the situation that you’re in!