While The Big Bang Theory never addressed the existence of George and Mandy's child, this could be explained by Young Sheldon season 6. Georgie hasn't had an easy time of it in Young Sheldon so far and, unfortunately, his appearance in The Big Bang Theory proves that things aren't going to look up for the supporting star any time soon. In The Big Bang Theory, Georgie admitted to Sheldon that he spent years trying to look after their mother Mary (as well as Sheldon and his sister Missy) in the years after George Sr's death, which was a major responsibility when Georgie already had a baby with Mandy.
However, like Meemaw's character shift between Young Sheldon and The Big Bang Theory, Georgie's child is never mentioned in the earlier series. In The Big Bang Theory, Georgie's failed marriage is addressed and this marriage might be between him and Mandy, the mother of his child. In the final scene of Young Sheldon season 6, episode 17, "A German Folk Song and an Actual Adult," Georgie proposed to Mandy, and it was left unclear whether she accepted his offer. However, characters in The Big Bang Theory didn't mention the fact that Georgie has a child, and Young Sheldon season 6 could explain why this was never addressed.
While Young Sheldon season 6 could reveal why Sheldon didn't mention that his brother Georgie is a father, it would be a bleak twist. If Mandy breaks up with Georgie over his proposal, this could explain why Sheldon forgot all about Georgie's child. Young Sheldon season 6's Missy plot is already making the sitcom's tone darker than it was in earlier seasons, and the story of Mary and George Sr's marital struggles isn't lightening up the atmosphere in the Cooper home. As such, breaking up Mandy and Georgie could be too dark and sad a twist for The Big Bang Theory's spinoff.
At the end of "A German Folk Song and an Actual Adult," Georgie told Mandy that he broke up with his girlfriend before he popped the question. However, viewers pointedly didn't see this breakup happen. If Georgie was hedging his bets and waiting until Mandy said yes before breaking up with his girlfriend, this could come to light and cause a breakup. While Young Sheldon season 6 has been hardest on Sheldon, the outing hasn't exactly been easy on Georgie either since his earlier attempts to woo Mandy ended in failure. As a result, Georgie may not have really broken up with his girlfriend and Mandy could find out about this.
While Mandy might simply not accept Georgie's proposal and this could cause a breakup, this would likely be a relatively amicable parting of ways. Thus far, Georgie has understood why Mandy isn't interested in a relationship, and he clearly adores their baby which means he wouldn't do anything to jeopardize their living situation. However, if Mandy discovered that Georgie was still secretly dating someone else when he proposed to her, this could result in a much uglier blowup between the pair. This could end with Mandy and their baby leaving Georgie's life, which would justify Sheldon never mentioning Georgie's child in The Big Bang Theory.
While Missy's fate after Young Sheldon wasn't as bad as it might have been, both Sheldon and Mary took her pregnancies as an opportunity to criticize her life choices. Missy's pregnancies and children are frequently commented on by Sheldon whenever she is mentioned, which makes the total erasure of Mandy and Georgie's baby a little strange in retrospect. The cause of this plot hole is easy to discern, since the writers of The Big Bang Theory didn't know that the series would spawn a spinoff and Georgie's past would eventually become a major storyline. However, in-universe, it is tough to justify this.
That said, if Georgie is still seeing his girlfriend after proposing to Mandy, this could result in him losing his baby and his love interest. This could explain why he leaned into caring for his mother and siblings after George Sr's death since Georgie may feel bad about his choices and want to improve himself. Much like Young Sheldon season 6 justified Sheldon's saddest traits, the series could also explain the erasure of Georgie's child. However, this would mean a sad Young Sheldon twist that may be too tragic for The Big Bang Theory's spinoff to pull off.