It's been quite a few years, but when I travelled to Europe with a six year old, it was not the airlines but the countries that required the proof. It may well have changed, but I would start my search with what the UK requires and maybe the US on re-entry. Further down on the page you cite it says contact the embassy or consulate of the country you will be visiting and ask about entry and exit requirements for that country, I think that's good advice.