This doesn’t happen if I have a bad cold , my nose will still run and I still cough while trying to sleep.
I can relate. The exact same thing happens to me, and I’ve often wondered about it.
Yeah, I’ve had times when coughing kept me up all night. But I’ve had many, many more times when I’ve dropped off to sleep, slept well and woken up naturally, then lain there and observed the chest irritation grow from nothing to cough-creating and the nasal effluvient start as a trickle and grow into a torrent.
But I have no answer to your original question. Sorry.
I read some discussion in or about some thriller/crime novel that said - if you duct tape over a person’s mouth, their plugged nose will clear soon enough to allow them to breathe. People who are gagged don’t choke to death even if they are crying in a way that usually means a plugged nose. Somehow your body senses the need to open the gates.
Never tested this, but feel free to let me know if any of you have any empirical observation to back this up… 
My observation is that if I sleep with a plugged nose, I will just wake up with a very dry mouth.
Talking about excretory functions–on very rare occasions I dream vividly about either peeing or pooping in bed. But it doesn’t seem like a dream at the time. A few times I’ve woken up absolutely convinced that I did, in fact, do whatever I was dreaming about. So far, however, I never have actually done so.
Does this ever happen to anybody else?
I highly doubt that because a lot of crime novels/ TV Shows have hostages suffocating from a duct tape over their mouth.
Now, there miiiiggght be a grain of truth there - not a Throat-nose-ear doctor, though: apparently, smelling and breathing is a hard Job for the nose, so every ca. 4 hours, one nose hole swells shut to rest, while the other nose hole does the Job, and then it Switches. Usually, you don’t notice if part of your nose is shut, because you can still breathe and smell (and there’s a lot of more important things going on), but if you have to sit still and concentrate on your breathing through the nose - maybe you suddenly feel one nose hole un-swelling and think in relief “My Body is helping me survive”.?
Yeah, Wild guess.