If the babies diaper is just wet and the baby doesn't wake up on it's own because of the wet diaper then there's no need to wake the baby to change the wet diaper.
The wet diaper will be okay on the baby until they wake up in the morning as long as it's not over saturated and leaking.
A diaper with Poop I would personally wake the baby up to clean them up and put them in a fresh diaper.
Not only will the poop cause a rash you likely wouldn't be able to sleep anyway if the baby is in your room since the poop will stink up the room.
I'm assuming the baby is in your room since the baby should sleep in the parents room for at least the first 6 months of the babies life but it's recommended to have the baby sleep in the parents room until 1 year of age to prevent SIDS.