this is complete nonsense.

the prime minister is not a separate executive office in canada the way that it is in the united states, so the question of "who becomes prime minister if trudeau dies?" is largely incoherent, in form. the prime minister is really just exactly that - the lead minister in the government, which means the leader of the sitting ruling party.

so, there would not be a process to find a new prime minister, there would be a process to find a new leader of the liberal party. this would be a temporary, interim position, due to the inability to launch a leadership process on short notice.

the governor-general would be then called on to invite the leader of the ruling party to form a new government, if there is confidence that they can pass a budget. but, the governor general does not make the choice as to who the leader of the party is, the party does.

so, the first thing that would need to be done would be for the party to name an interim leader, and the second thing that would need to be done would be for the governor-general to invite that interim leader to form a government, if there is confidence that they can pass a budget.

so, would this be a temporary or interim position, then? well, it would be relative to an upcoming leadership process. it's not the unwritten constitutional principles that determine the longevity of the position, it's the internal political processes within the parties, themselves. i suppose that the party might choose to forego a leadership process, or that the elevated candidate may run opposed, but a cabinet member cannot somehow inherit or be donned with party leadership, there needs to be an actual vote.

and, i will repeat that it would be widely expected that anybody that is intending to run for the leadership after the interim period would step down during the selection process, so as to not exert undue influence over it.

https://globalnews.ca/news/6790626/coronavirus-trudeau-prime-minister-sick/