Quite often the father named his natural or illegitimate child in his will, as often did one or more of the grandparents, or sometimes a sibling. If you read through transcribed wills you will see numerous examples of this.
When James Christian d.8 March 1729 he mentioned in his will "his sister Margt Dowan and her son John Sayle". In 1723 Thomas Sayle had remembered this son "to his illegitimate son John Sayle" (I didn't fully transcribe these wills because I saw them pre-Brian Lawson's wills website). However, see Thomas Creer 1754 "to his natural son John he left five shills legacy"; Henry Skillicorn 1835 "Likewise we Leave to John Skillicorn Illigitimate son to our said son John Skillicorn Two pounds British..", etc. To find these you need to look extensively through likely wills films at a LDS Family History Centre, or MM.
Sometimes you can find him/her in a census living with a paternal grandparent or other paternal relative, especially if the mother has died.

Sue