That trip down memory lane would reassure me that I'm on solid ground to give the expert as much time off as he wants and to go to the boss with a request for an extension on the project or some other fix. Company policy says that employees in need of personal time for family crises should be granted that time; and bosses understand that shit happens and they should have flexibility to deal with that.
Aside from requesting a deadline extension, my review of past projects would also suggest two other possible fixes: Ask the boss for another expert to be assigned to the project from outside, or replace the expert with his own assistant.
Based on how he keeps asking for updates, it sounds like the boss is pretty hot to get the project done on time; so I would explore the replacement options first.
Bringing a second expert aboard the project would probably be the easiest and least disruptive. But the fact that the inexperienced assistant has volunteered to replace the expert would weigh heavily in favor of using the assistant: I like to reward initiative whenever possible. So even though using the assistant might introduce more variables into the equation and lead to more headaches in the long run, I would explore that option first.
Naturally I would want to cover all the logistics of that arrangement. For example, since the assistant is inexperienced I would need for someone to review his work (the original expert, me, another team member, or an outside expert). I would probably also need another person to do the assistant's original job.
But assuming that all logistics are doable, the last step would be to make my case to the boss and the team members and get them to buy in on that particular solution. The boss would probably be satisfied with the argument that training assistants to become new experts is an ongoing process, even during projects, and there should be duplication of knowledge and experience so that the team isn't crippled when one member has an emergency. The team members should be satisfied with the argument that using the assistant will ensure the greatest continuity in the project to the extent that the assistant knows the project and knows the original expert's methods. If the original expert doesn't like being pushed aside, I would dicker with him to keep him involved on a part-time basis and give him credit for overseeing and training the assistant to a new level of competency.
It isn't strictly necessary to come up with reasons for the team members and the original expert to buy in on the proposed solution; I'm comfortable with ruling by fiat and simply ordering people to do things my way. But team members are stakeholders too; and high-pressure projects are a special environment that may require a high degree of consensus within the team; and assuming the team members are good at what they do and they know the expert and his assistant, then they may have some important considerations of their own to contribute to my decision-making process. So I would probably want to get the team members on board with this decision too. And the goodwill earned from the buy-in process is a bargaining chip that can be traded in later when the going gets rough.