updated 07/11/2011 AT 1:00 PM ET
•originally published 07/11/2011 AT 2:10 PM ET
Luckily, they have a bit more time to decide – although the actress, 32, and rocker, 33, plan to settle in London while Bellamy’s band Muse works on their next album, they’ll remain in Los Angeles while Hudson recovers from the birth.
The Something Borrowed star – who delivered her 7 lbs., 12 oz. son naturally despite having undergone a Cesarean section with Ryder Russell, 7½ – is “doing great,” says the source, adding that the couple is “beyond thrilled” with their new addition.