The Foreign Minister Dominic Raab is standing in for now. If Johnson dies, or is forced to resign it is a bit unclear how a successor would be named on a more permanent basis. The PM is the leader of his/her party in the Commons. Normally there would be a party election while the outgoing PM remained on until a successor was confirmed. My guess is that in a worst case scenario, the Queen would formally ask Mr. Raab (with the consent of the cabinet) to serve as acting PM for the duration of the emergency.
Thank you.