Ok I've spent countless hours trying to sort this out on my Note II. I tried everything suggested, nothing worked.
Until I went in to a conversation and tapped on the drop down box where the number is displayed top left. It had an option "Add to people". That alllows you to save that number to a contact. I noticed that the number that Hangouts auto inserted was the same number but with the country code. Instead of saving a duplicate entry though, I deleted the one Hangouts inserted and just added the +44 to the original number for the contact. When I saved this, the contacts image displays correctly. I updated several other numbers with the country code and now their images display.
So the problem seems to be that Hangouts won't recognise numbers as linked to a contact without the country code.
Update: ok adding the country code doesn't seem to solve the problem. What does is tapping the "add to people" and choosing a contact. You don't need to complete the adding of the number, just initiate the process.