The other, Finding Your Feet, works to provide physical and emotional support to amputees and has transformed its operations to work online, running counselling, exercise classes and virtual meet-ups.
The Duke, who called each charity on Zoom, praised them as "inspirations" and congratulating them on helping others at an "extremely busy" time.
The NET, of which he is patron, has already distributed £32 million ($64m) raised through its Coronavirus Appeal, helping 5000 grassroots charities to adapt to meet the urgent needs of their communities.
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In conversation with Finding Your Feet, he bonded with its founder and a volunteer over their shared love of motorbiking, and shared his belief that the Paralympics had "completely changed" public attitudes to seeing those who had lost limbs.
At Peek, he heard of the importance of providing nutritious meals to families who are missing free school meals, and how the charity used the NET grant to source toothpaste, sanitary protection, nappies and baby formula for families who did not have the resources to bulk buy.
Given a tour of Peekachew - named by the children it helps - he heard from community chef Charlie Farrally about their record 900 meals made in one day in the van and kitchens loaned by local businesses, saying his efforts were "incredible".
Told by Farrally the meals were all cooked from scratch with "quite a lot of love", he heard they were "taking the pressure off parents" who are already coping with the effects of the coronavirus lockdown for children with sometimes complex needs.
"You'll know yourself, the hardest time is dinner time," Farrally said.
The Duke, a father-of-three, laughed and replied: "It depends what's on the table. If parents put something on that children love, dinner time goes very well. But if you put something on the table they don't want to do, that's another ball game."
He also spoke to Michaela Collins, who started visiting Peek as a service at the age of 9 before becoming a volunteer and rising to the position of chief executive.
"What a brilliant rise," said William. "I think that's absolutely fantastic. You are the embodiment of what can be done."
Asking about the particular challenges faced by the people Peek help, he heard that single-parent families, those with new babies, refugees, those who normally receive free school meals and those without disposable income for a weekly big shop, had been helped with food and basic health provision.
The Duke, who was appointed Lord High Commissioner to the General Assembly of the Church of Scotland this year but could not visit this week in person as planned, added: "I hope when I find myself in Glasgow in the near future I can come and see you guys in person and congratulate you."
Collins said they had offered the Duke cooking lessons, as well as a "Team Peek" hoodie.
In a second call, to Finding Your Feet, the Duke met founder Corinne Hutton, who was given a 5 per cent chance of surviving sepsis in 2013 and had her hands and legs amputated before the knee.
After setting up the charity to help others, she has become the first quadruple amputee to climb Ben Nevis, the first female quadruple amputee to climb Mt Kilimanjaro and the first Scot to receive a double hand transplant.
The Duke told her: "You're a huge inspiration. I'm so pleased the NET have found you and you've found them and that relationship is helping."
They were joined by Stephen McAtamany, 57, who had his left leg amputated below the knee in 1988 after a motorbike crash, and now sports a leopard-print limb.
Asking whether it had been a challenge to swap their usual real-life meet-ups for socially distanced online versions, the Duke heard how exercise classes, counselling and quizzes had moved quickly online thanks to an emergency grant from the NET.
McAtamany spoke candidly about his own mental health, which had left him suicidal at one point before he reached the Finding Your Feet counsellor he credits with saving his life. He is now a volunteer for the service.
"What was the biggest bit of support for you?" the Duke asked. "One of the things we're trying to encourage people to do is have that first conversation. For you, that first conversation with Michelle [the counsellor], did those dark clouds lift a bit?"
"Being able to reach out initially to make that appointment was a massive step," said McAtamany. "A big burden was lifted right away."
Of the challenges facing new amputees, whom the charity tries to reach as early as possible, the Duke said: "Individuals react in all sorts of ways don't they? You never know how people are going to react to that bad news when they wake up and realise they're missing a limb.
"It's something I've tried to understand and imagine: some of the guys coming back from operations, how they go through that process and understand it. Your life must change in a second.
"Thank goodness Finding Your Feet is there to look after all those people who have suddenly woken up and realised their life is very different now."
He added: "One of the things I noticed after the Paralympic games was that the country's attitudes towards those who had lost their limbs was completely changed. Is that true? Do you guys see that and feel that as well?"
"Definitely. In the old days you'd wear your trousers," said McAtamany. "Nowadays I've got a leopard print [leg] and it's just out there when you're wearing shorts in the summer. The more people see me and Corinne being open about it, the easier it is for others. People don't look twice now."
The calls were set up at the end of last week, taking place via Zoom on Tuesday.
Hutton, 50, said she had been "thrilled to bits" to learn Prince William wanted to hear more about Finding Your Feet, adding: "You never normally get to say thank you to the people who help raise the money. It was a super opportunity."
"He'd obviously done his homework," she said. "He was very complimentary about the charity and my role in it - he made me blush a few times. He really cared."
Saying the charity had made more than 1100 calls to people at home already, to check "nobody's slipping through the net" in lockdown, she added that the emergency fund had been essential in allowing them to keep providing online services for those that need it.
"I love this charity and the people in it, and it's lovely to speak to someone who's interested and hear them pick up on that," she said. "This is a member of the royal family and you don't get any higher than that."
McAtamany, who joined Finding Your Feet after seeing a leaflet in his limb-fitting clinic, said he has found many friends as well as receiving life-changing counselling, and has now become a volunteer to give something back.
He said he had been "very honest" with the Duke about his own mental health, being at a "really low ebb" before finding help.
"I was really nervous but he's such a gentleman," he said. "He obviously knew a bit about us too.
"The fact that he knew about us and was interested enough to speak to us, I think it's absolutely wonderful. The fact he addressed the mental health stuff was really good as well.
"I lost my foot because of a motorbike accident. He was telling me he was a biker too, he loved it too, and he was asking if I still ride.
"Nothing was put on at all - he was really interested, really grounded."
Collins, 29 and the CEO of Peek, said the NET fund had allowed them to support those with no disposable income, from keeping children who would otherwise receive free school meals fed to shopping for nappies for newborn babies.
Having just picked up the keys to the food van shortly before the Covid-19 crisis hit, she said she speed of the funding had been key in getting help to those who needed it quickly.
"There's such a stigma attached to poverty," she said. "People in desperate need may not ask for help because of the pride factor. But through Peek we have that trusted relationship: it's not a hand out, its a hand up.
"The public response has been heartening.
"People are appreciating that it can happen to anyone. That's what's bringing people together.
"We're all affected and we need to support each other through this with kindness, caring, compassion and empathy."
Of the Duke, she added: "He seemed [to] genuinely really care about what's happening across the UK - that whole sense of community spirit and helping each other out. It's difficult for everybody but particular families impacted by poverty on a regular basis.
"What's so nice is that it was a genuine conversation and he really wants to reach out to show support. It was really nice and certainly was a morale boost."
Asked about their discussions about the importance of healthy eating, she added: "We did offer him a cooking lesson so hopefully that will be something we can do in the future."
