Monarch's powers in appointing the prime minister

Hi everyone!
Happy holidays to you all! After the recent lengthy Tory leadership
election, I have been wondering what would happen if such a contest
were to occur (in the UK) if the party were in government. More
specifically, who would be prime minister in the interim?
Here's a hypothetical:
The Prime Minister dies/resigns mid-way through a parliamentary
term. I am aware that convention has it that the Queen would chose the
new leader the governing party elects as the next prime minister.
However, what if many candidates put their name down in the leadership
contest? Surely it wouldn't be possible for the country to go without a
PM for over a month while the party whittled down the candidates to
just one?
I suppose what I'm asking is, in these circumstances, would the Queen
select a temporary PM from the Cabinet simply based on her own beliefs
on who would make a good interim PM? Or is there some old convention
that she would chose the party chairman or Home Secretary or Lord
Chancellor or something? I know some precedents exist (Alec
Douglas-Home?) but this I believe was before the Conservatives extended
the vote for a leader to MPs and party members.
Or would the office of Prime Minister just be left vacant until a
new leader was elected (for potentially several months??) since
presumably if the Queen selected a PM and then another MP were elected
leader a few weeks later and the ruling PM didn't resign he would have
to be dismissed by the Queen in order for the newly-elected leader to
take office?

Don Aitken...

The PM remains in office until a successor is appointed. This
situation has happened twice since the parties started electing their
leaders. Wilson, in 1976, resigned the leadership of the Labour Party,
but remained Prime Minister while the election was conducted,
resigning that office only when Callaghan had been installed as
leader. Thatcher, in 1990, resigned from nothing, remaining both
leader and Prime Minister until her successor was chosen.

(Gary Holtzman) garyholtzman...

But what happens if the PM dies?

Sorry if this has been a bit convoluted! I'd be very interested to
hear your thoughts!
Best wishes,
Jonathan