The answer to the connection to the queen is the next line.
"British MI6 agents dead"
That's how Q works. The following line references or answers or guides the previous line.
So it would be:
London Mayor - Sadiq Khan
Background - Pakistani Sunni Muslim family
Affiliations - Extremist Muslims, David Cameron went as far to claim he had ties to ISIS. Rumor of ties to Al-Qaeda, Hamas, MB.
Connection to Queen? - Q is implying that the connection is the British MI6 agent that was killed: Gareth Williams, a mathematician, cryptographer, who was supposedly involved in counter-cyberterrorism
So how was Khan connected to Gareth Williams, and how was Gareth Williams connected to the Queen?
Q goes on to guide the line of questioning for the answer:
When? August 16 2010
How? Murdered and placed in a duffel bag padlocked from the outside.
What was reported? It was an accident
What really happened? It was murder.
Why is this relevant? In the context of this line of questions: Sadiq Khan related? Did he find out proof of SK's radicalism? Did the Queen order his death?
Is Sadiq Khan the Old Connection to Diana? He's a bad actor? Can we find proof of this in the news of her accident/death?
What was Sadiq Khan doing back then?
Why talk about Diana now?
It's old news
but there's a connection to be found.
Read the news about the incident.
Find the Bad Actor.
Bad Actor = London Mayor.
I believe Q is implying that Sadiq Khan was involved in Diana's murder, and Gareth Williams found out some how through Sadiq Khan's terrorism connections (his field of work). He was shoaed to cover up.