Nothing, technically, prevents a God from taking up residence in the Underworld. Like, if his domain is destroyed or taken over by another God and he has a pack of Immaculates out for his permanent dissolution then he may very well end up hanging out in the Underworld at the place analog to his old domain. It would be a absolute last resort because while there he would not respire Essence at all, be unable to create a Sanctum, have no defenses against the world other than his personal puissance and probably attract the attention of the Deathlords.
Basically, running to the Underworld when your domain is destroyed is like running to Syria when you lose your job. Far more likely the god in question would try to sneak in during the Carnival of Meeting and then go on the dole with all the other out of work gods, accept a demotion to a less powerful position or even burrow deep into the Earth and hang around with the forbidden gods than he would ever consider going to the Underworld.
Once in the Underworld, however, the God could get himself some prayer from Ghosts, maybe even attune an Underworld Manse for some mote regen down there. If he works out some deals with local powers he may be able to set himself up all right, depending on exactly what he brings to the bargaining table. If he’s desperate enough, I’m certain there are Necromancy spells and other rites of the Underworld available from the Dual Monarchy that could get him converted into a Creature of Death so he can start respiring down there as well. This is a bargain he is very likely to regret in the long run.