I cant believe this OP.
The Queen does not run anything, and everything she does do is a formality.
She can appoint ministers, but so can the Prime Minister so if the royals all died that wouldn't stop.
She can issue Royal Pardons to criminals, but she doesnt really do that.
She does need to sign laws in order for them to come into effect, however it's not an issue. The last time a monarch refused to sign a law was in 1708. The royals dont use their influence on that; they just sign what they're given.
She appoints archbishops, but she's just told who to appoint. She doesnt decide.
She can declare war, but needs government approval. She never has.
She can decide dissolve parliament and force new elections. She never has.
She technically commands the armed forces but in reality, she doesnt do anything. If the whole line of succession died, armed forces would carry on. She can also name someone else the commander-in-chief if she is very very I'll, so she would likely name the PM.
She also bestows honours. Not really vital to the running of the country.
Everything she does is just.... for show. The few powers she does have dont actually affect the running of the country because she doesnt use them. No Royal would now. Unless the government turned all Nazi or something.