Hi
As a recently retired immigration officer I hope the following information helps, to travel within the EU you can travel out from the UK and return on the very day you passport expires, although this is something I would not recommend, because if your return is delayed for whatever reason, you might have problems leaving that country, as you would techniciacl be undocumented, and you could be told to go to a consulate to get a valid document.
Also technically to be in another country without a valid ID document, in the case of a Brit thats a British passport, then you could be fined or worse arrested as an un-documented alian.
With regard to the USA you passport must be valid for duration of your stay, but again for the reasons I have already given, this would not be a wise move. Australia & New Zealand have their own requirements including visas etc, so you are better of going to that countries websites.
As for returning to the UK without a valid passport, you might be asked to explain why you travelled knowing your passport has or might expire when you are out of the country, checks would be made to ensure that the passsport was in fact yours and had not been lost, stolen or counterfiet.
Steve