Simon Cowell Baby Mama Lauren Silverman Flashes Bump as Dad-to-Be Takes Time to "Adjust to Unexpected News"

Soon-to-be divorcée carrying the X Factor judge's child steps out in the Hamptons with her 7-year-old son

Talk about going from everyday soon-to-be divorcée to woman of international mass fixation overnight!

Lauren Silverman, who is pregnant with Simon Cowell's child, was spotted out in the Hamptons on Thursday, both with her 7-year-old son and alone, a hint of a baby bump starting to show underneath her clothes.

This was the first sighting of Silverman, who is still in the middle of divorcing her husband (and Cowell pal), Andrew Silverman, since the news that made her insta-famous broke yesterday.

But while obviously the news came as an enormous surprise to, well, pretty much everyone in the world who has heard of Simon Cowell, just how surprising was the news for the expectant father himself (whenever he was clued in, that is)?

A source exclusively tells E! News that the 53-year-old bachelor seemed close to cooling things off with Silverman just last week, expressing his reluctance to be "tied down."

Team Cowell is looking at Monday or Tuesday for a possible press conference and in the meantime he is "taking some time to adjust to this unexpected news," the source says, adding that the Brit is currently focused on meeting up with the reigning No. 1 lady in his life—his mother, Julie.

We hear that Cowell, who at least was scheduled to appear at Fox's Television Critics Association panel this afternoon in L.A. (we'll keep you posted), is being advised in the meantime not to make any TV or radio appearances and to stay off social media. His exes have been told to lay off the tweeting and blogging as well, we're told.

Across the pond, the U.K. X Factor team, including judge Sharon Osbourne, have pulled the plug on promotional opportunties "until further notice," the source says.