After editing any of the details, press enter to confirm the edit, before pressing 'Update User'.
This to make sure you need the changes.

Doesn't work for all fields.

When I get to Billing address there are 6 fields (4 text and 2 drop downs) for the one section.

Pressing Enter won't save any of them. (and how can you do that for a drop down anyway?)

Surely this can't be how it was designed?

Sorry, that works for text fields. For all fields, there is an icon on the right. When it's X it means clear this attribute value.
When you click on the attribute title to edit, it turns into a notepad with a pen.
When you change the value of an attribute and want to save it, click on that icon after your update, ** EDIT ** It will be applied automatically. No need to click on 'Update User' when you're done.
Hope this is clear.

Still doesn't work properly.

If I select the country as Australia and the State as Western Australia it records it as Washington, Australia, even thought the drop down is clearly showing Western Australia.

That's a serious bug in the program.