Generally speaking the main difference depends on whether you are a returning to a country you are a citizen or - at the very least - permanent resident of or not. If not, chances are slim to none that any airline will allow you to board.

If you are returning to your country, then the answer is slightly more complex. The passenger in this scenario has cleared immigration (the UK doesn't do exit immigration but the passenger is non the less airside). So the airline document check essentially boils down to knowing if everyone will be let in at the destination. If you're a national of that country you may be able to produce other document that may serve as proof of citizenship/residence even though they wouldn't be considered travel documents (e.g. ids issued for domestic use only).

That said, in most cases airlines will chose the easiest way to the detriment of the passenger and refuse boarding. That in turn raises the question of what's going to happen to the passenger that is neither a resident nor a citizen of the country he's departing from.

As far as the immigration is concerned the passenger has already left the country and to obtain a new passport from the local embassy/consult would need to reentry that country which won't be possible without a passport. Will the passenger be put into a holding cell and ultimately be deported once his identity has been confirmed? Or are they just going to rehash Tom Hanks's terminal?