The queen was seen arriving back at Buckingham Palace from Windsor Castle, just outside the capital, in mid-afternoon but other royals went about their usual business.
Charles, the current heir, was visiting York in northern England, where members of the public shouted "Congratulations!".
Smiling, he replied: "Do you know something I don't?"
Charles, who turns 65 in November, joked: "I'm very grateful indeed for the kind wishes for my rather slowly approaching grandfatherhood."
Prime Minister David Cameron sent his best wishes to the couple and the "whole country is excited."
The pregnancy was announced in December when Kate was admitted to hospital with severe morning sickness.
At the Lindo Wing, a standard room and normal delivery -- which Kate is hoping for -- costs £4,965 ($7,600, 5,800 euros) for the first 24 hours, plus consultants' fees which can reach around £6,000.
The duchess is being tended by a top medical team led by the queen's gynecologist Alan Farthing and his predecessor Marcus Setchell.