Here are a couple of verses that I transcribed from two different sound samples; therefore I don't know whether there's more, or whether I have them in the right order.

Walk on, little boy. Your life has only started.
A world full of joy is there for you to see.
Walk on, little boy. Our paths today have parted,
So walk along, run along, but don't you wait for me.

Walk on, little boy. We've been too long together.
Don't cry, little boy. It's time I set you free.
Walk on, little boy. You've seen the change of weather,
So walk along, run along, but don't you wait for me.

allmusic says it was written by "Gilkyson"—that's probably Terry Gilkyson.

MusicStack says it was also recorded by Arthur Godfrey!