Given as Caroline's office didn't make an announcement about the birth of her first grandchild, I can't see that happening with the birth of her second grandchild.
That said, both Charlotte and Gad are public figures because, regardless of Charlotte's family connections, both have pursued career paths that make them celebrities. Charlotte is a celebrity who is known for keeping her personal life more private than others, while Gad is not. Therefore we can likely expect to hear something about the birth of the child through Gad at some point - much like how we would eventually hear about the birth of any other celebrity's child.
Princess Caroline did put out a statement from her office at the Palace announcing the birth of her grandchild.
Princess Caroline’s press office confirmed the news via a statement. “Her Serene Highness Princess Caroline has the pleasure of announcing the birth of her grandson, born on 21 March 2013,” the missive said before adding: “Both mother and baby are doing well”.