In the game Two Rooms and a Boom, the leader of a room is the first person to be nominated by another player. What happens if no one has done this in a room by the end of round one and the room has no leader?
The direct answer to your question is there isn't a rule that covers this eventuality. The rules state however:
In Two Rooms and a Boom, part of the fun is the lack of rules. Players can say whatever they want, do whatever they want, and come up with whatever strategy they want.
I think it would be a fairly easy to say a leader must be chosen before people swap between rooms. Otherwise you could ask the question what if someone hasn't been chosen by a leader before rooms must swap? The answer is the game doesn't work. You have to send someone across and so you also have to have a leader by that point.
If your group is struggling in particular with this then maybe you need to appoint a Games Master for the game for a few playthroughs until people get into the swing of it?
We play this game almost every week for multiple rounds at the University game society. We have never had this problem. (In fact it is rare that leader isn't decided within a few seconds of a round starting as nearly always the first question asked is "Who should be leader?")