I still maintain she answers the phone:
“Queen here, who’s there?”
Or (in subzero British tones) “William.”
Congratulations to all involved! Very neat.
I’ll add my congratulations. It was a great question, Kimmy; and an informative column, Elendil’s Heir!
“Halloo, ‘Get more cash from K-ROX, number one all-rock station in the Las Vegas area,’ this is the Queen speaking…”
Many thanks to Kimmy for asking the question in the first place, to Northern Piper for his help with the Canadian DJ, and to Ed Zotti, TubaDiva and C K Dexter Haven for helping bring the column to publication. I’m much obliged to you all.
My compliments to Elendil’s Heir, it is a thorough report! I suppose my hopes (and Polycarp’s) that it would have been “Oi, this is Brenda!” was probably a little too glib … but good on the princes for recording “Wassup, this is Liz.” 
Bumped.
Here’s more on the Queen’s sense of humor: Queen Elizabeth II's Funniest Quotes | POPSUGAR Celebrity
She has, or had, occasional lunches for a mixture of people she wouldn’t otherwise meet informally. I’ve seen it reported that one guest was a trades union leader with a bit of a firebrand reputation, and on that day it was in the papers that he’d had to settle a dispute on worse terms than he’d been holding out for; at the lunch, a particularly crispy piece of roast potato escaped his fork and shot off the table, only for a corgi to waddle over to it, sniff at it and walk away without touching it. And the Queen said “Oh dear, Mr… - not your day, is it?”
And one that was in a TV documentary about her working year: at a reception for a Commonwealth Heads of Government meeting, not long after the first Iraq war (after the invasion of Kuwait), she was with a group of assorted politicians that included Edward Heath, the former Prime Minister and famous sulker at being ejected years earlier in favour of Margaret Thatcher. He was holding forth in his usual rather pompous manner about the run-up to the war when Saddam Hussein was holding a number of British people more or less hostage, and Heath had gone to Baghdad to try and get him to release them: “I was the only one who went, you know, no-one else would go”. “Oh,” laughed the Queen merrily, “but you were dispensable”. You can imagine his face…
When Martin McGuinness, Sinn Fein politician and former IRA commander, met the Queen it was a very big deal.
It was widely reported at the time that the formal dinner included ice cream bombe. Hard to imagine Her Maj not having a chuckle at that as she approved the menu.
Personal-ish anecdote.
My parents have met the Queen on a couple of occasions (mainly because they work in the equestrian world). At one drinks reception, they witnessed one of their friends rush into the room and almost barge into the Queen. In a flustered state, she mumbled ‘Oh, sorry Ma’am, I’ve lost my husband’, to which the Queen replied ‘lucky you!’.
This is the conversation with the Quebec radio DJ pretending to be the Canadian prime minister:
The Queen does, in fact, pick up the phone and say “hello” like anybody else. The fake PM says “hello” in return, after which she says “Ah, prime minister”. No special protocol, just a normal telephone conversation.
Well, it does mean that she has already been told who is on the line. Which in turn means that she doesn’t need to tell the caller who she is, because the caller has already been told that he is being put through to her.
This stands out as the best answer IMO.
It’s formal, but not unnecessarily involved.
It strikes me as old fashioned, and appropriate like a personal introduction.
And it seems I’ve heard this on old TV shows.
Like maybe, Bewitched. Think of an important client calling Larry or Darren.
Bumped, as the column is back on the Straight Dope homepage: How does the Queen answer the phone? - The Straight Dope
Shouldn’t that be “This is My Majesty”? (Or “This is Our Majesty” when she’s feeling dynamic.)
“Oi, ews 'is then?”
Moderator Action
Since there is a Staff Report on this now (which did not exist when this thread was created), let’s move this from GQ to Comments on Cecil’s Columns/Staff Reports.
How many hundreds of times have we told newbys to include a link when their post refers to a published report?
And now an experienced moderator forgets to do so? I blame it on that Covid-19 virus.
The link is in Elendil’s Heir’s post, three posts up from mine. Seemed silly to post it again.