The centre is aimed at young military personnel who generally live in shared accomodation.
Oasis provides a space for those who need "me time" and also provides a place they can make private video calls.
William, a former Royal Air Force pilot, recalled having to use token so he could make a 15-minute phone call home.
“FaceTime makes such a difference now,” he said.
Kate also gave some rare insight into the couple's third child - 7-month-old Prince Louis.
Rhian Davis, 32, who was carrying 10-week-old daughter Dorothy, said the royal mum stopped to chat about their young children.
“She asked whether she was sleeping through the night, and she told me that Louis is starting to wave,” Davis told PEOPLE.
Kate also revealed she had tried to show her children, Prince George, 5, and Princess Charlotte, the treats the military families had made at Kensington Palace for yesterday's party. However, when the children arrived home from school and nursery, the treats were all gone.
“We took the children down to where the party was but everything was gone,” she said. “They were so disappointed!”
However, George was apparently already preoccupied with something else!
William said, “I told George this morning we were going to see the pilots today, and he said, ‘If you see a helicopter, can you take a picture?'”