If there is a single hole in the bucket as the song relates then I think two things stem from that -

firstly the hole must be in the bottom of the bucket otherwise it could contain some water, certainly sufficient to wet the sharpening stone

secondly that the water is to be drawn from a well, otherwise by tilting the bucket away from the hole you could again get some water in it - this would also negate the finger in the hole solution

We can therefore gather that there is no running water in the household, and no stream or lake in the vicinity - and that it isn't raining - I am getting visions of the American mid-west here - the lack of water does not suggest the UK

Henry (or Georgie for you Penny) is obviously laid back and I can quite picture him sitting in front of a wooden shack somewhere on the prairies while his spouse (is that too big an assumption to make) gets frustrated at his demeanour

They are probably not wealthy nor do they live near a town and are not on the internet - otherwise you might get -
There's a hole in my bucket dear Liza dear Liza etc
Then buy a new one dear Henry dear Henry

are the names Liza and Henry more popular in one region than another and do they give a clue to the ethnicity of the subjects?

That they own a blunt axe would indicate that they live near woodland and that either Henry or more likely Liza has been using it a great deal

can we glean any more from the song?