According to a source, baby Eric is "doing well, he's very strong. He can already hold his head up! He's not fussy and and he's a good sleeper."
Simon Cowell pictured with baby Eric
A friend of the couple also told E! that the new arrival has already met his half-brother Adam, from Lauren's marriage to property developer Andrew Silverman.
"Adam came to London for his spring break from school. He adores his little brother and loves helping with him," the source said. "It's a happy time. They're settling into their new life."
The couple are said to be due back in New York in April after Simon attends the opening of X Factor spoof musical I Can't Sing.
Simon recently spoke about his feelings about fatherhood: "I was born to be a dad," the X Factor judge told his ex and long-term friend, Terri Seymour, on Extra.
He also said that he was reduced to tears when he laid eyes on his son for the first time, and was surprised by the immediate love he had for the precious bundle of joy.
He said:"I actually said to Lauren beforehand, 'What if I like the dogs more than him?' and she said, 'You're gonna be absolutely fine,' and then literally within seconds of him being born, I got it!"Simon, whose baby news with girlfriend
Lauren Silverman came as a big surprise, seems to be a natural. Since Lauren gave birth to baby Eric – who the couple named after Simon's late father Eric Selig Phillip Cowell – he has posted several touching photographs of the family.
And there may be additions in the future, according to the music mogul. When asked if he would have more children, the 54-year-old replied: "I wouldn't rule [it] out."
He continued, mentioning his two treasured Yorkshire Terriers: "Well, I got Squiddly and Diddly, and I couldn't imagine Squiddly or just Diddly...I had a brother when I was younger, so I kind of think it's cool to have two or three... two."