Minchin, 53, teased that she was returning to present the show with Nugent, 50, after Walker recently announced he will soon leave the BBC after 13 years to a host Channel 5's 5pm news programme.
Walker, 45, shared a clip of the two women sitting on the sofa together with Minchin saying, "Just for old time's sake Sally go on."
— Dan Walker (@mrdanwalker) April 27, 2022
Nugent then addresses the camera: "Hello. Good Morning, and welcome to BBC Breakfast," before Walker popped up behind then saying: "Um, you're stealing my job?"
Minchin laughed: "Not for long Dan!"
Walker - who competed in Strictly Come Dancing last year - recently announced his departure from BBC Breakfast, which he has presented since 2016.
The former I’m A Celebrity contestant said: "For me it had physical implications, I had heart palpitations, tinnitus, all sorts of things, and this has a real impact at work.
"The brain fog is very real. I remember sitting where you are Sally, interviewing the then Chancellor George Osbourne and he was outside a JCB factory. And I couldn't say the letters 'JCB' because my brain just went into this extraordinary reeling.
"These are very real problems and other people in different kinds of work will have other issues. That was before I had HRT."
She added of starting the conversation about the menopause on BBC Breakfast: "I wasn't on HRT at the time, and actually I look back at the film that I did and I kind of feel sorry for that Louise.
"It looks to me like a different person because I was so anxious, I was so shaken up."
Current shortages of HRT in the UK - attributed to manufacturing and supply issues - are leaving women battling withdrawal symptoms including hot flushes, night sweats, low mood, anxiety, joint pains, vaginal dryness and discomfort during sex and difficulty sleeping.
Minchin said: "I think it's important to keep the momentum going, get it back, it is a supply problem it must be able to be solved somehow."
Watch: Dan Walker gets emotional as he announced BBC Breakfast exit