Princess Kate and Prince William 'Enjoying Family Time' at Home with the Kids

In their first days as parents of two, Prince William and Princess Kate have been quietly bonding with their children at home like any other family welcoming a new baby – except home is a palace and a visit from great-granny means a pop-in by the Queen herself.

Since arriving home on May 2 just hours after Princess Charlotte’s birth, the new family of four “have been enjoying nice, private family time,” a close palace source tells PEOPLE in this week’s cover story.

Prince William Prepares for Fatherhood

Prince William may not let his six-months-pregnant wife, Kate, win at ping-pong – “Ahem, I think that was 6-2,” he teased after beating her in an impromptu match at a Scotland youth center April 4 – but he is getting ready to make bigger concessions to his family-in-the-making.

Prince William and Kate Middleton Are Practically Family

They may not be engaged yet – but Prince William and longtime girlfriend Kate Middleton already are practically family.

During a recent lunch at a ski resort in the French Alps, an observer witnessed William and Middleton sharing a very chummy lunch with her parents. Says the onlooker: “William was clearly very close to Kate’s father, and every time he spoke to him, William replied, ‘Yes, Dad.’ ”

So, when will an engagement happen?