Some newspaper announcements that I can find:
"Tuesday the Lady of Lord North was safely delivered of a Son and Heir."
September 13, 1757 - September 15, 1757
Publication: London Evening Post
"Saturday the Lady of the Right Hon. Lord North, one of the Lords of the Treasury, was safely delivered of a Daughter at his House in Grosvenor-street."
February 16, 1760 - February 19, 1760
Publication: Whitehall Evening Post
"Saturday the Lady of Lord North was safely delivered of a Son, at his House in Grosvenor-street."
December 26, 1761 - December 29, 1761
Publication: Whitehall Evening Post
There is also this one:
"May 31. At London, the Lady of Lord North, of a son."
Thursday, May 1, 1777
Publication: The Scots Magazine
Will keep looking for other birth announcements, however, note the one in 1757.