"We're told that Eugenie and Jack are doing really well," royal expert Katie Nicholl tells ET. "They are thrilled to be parents. They are absolutely overcome with love and emotion for their little son."
The couple welcomed their baby boy on Feb. 9, and revealed his special name earlier this week -- August Philip Hawke Brooksbank.
"It was very special that [Prince] Phillip -- of course, Eugenie’s grandfather to whom she's incredibly close -- has been honored and [paid] tribute to in the naming of their child," Nicholl shares.
She also says of the baby's unique first name, "August is an unusual first name [as] he's not an August baby. But apparently it was the name that the couple really like."
According to the royal expert, Princess Eugenie -- the daughter of Prince Andrew and Sarah "Fergie" Ferguson -- and her husband "tend to be very modern when it comes to parenting," so they've thus far decided to forgo a royal title for their newborn son.
"They don't want that burden of a royal title [on] their son," Nicholl says. "I think this little royal is going to grow up an ordinary little boy indeed."
Eugenie gave birth to her baby boy at The Portland Hospital in London, Buckingham Palace confirmed in a statement at the time. Notably, it is the same hospital Prince Harry and Meghan Markle welcomed their baby boy, Archie, in May 2019.
While the new parents are spending time at home with their baby boy, Princess Eugenie's 99-year-old grandfather is currently in the hospital in London, on the advice of his doctor.
A royal source previously told ET that following a consultation with his doctor, the Duke of Edinburgh "is likely to remain in hospital for observation and rest over the weekend and into next week."
"The doctor is acting with an abundance of caution. The Duke remains in good spirits," the source added.