My mum and gran always sang it like this:

Felix kept on walking, kept on walking still
With his hands behind his back, you would always find him
Blow him up with dynamite, but him they could not kill
Through the air Felix flew
He just shouted "Toodle-oo!"
Landed down in Timbuctoo
but he kept on walking still.

Felix had a baby, had a baby boy.
and he called him Paddy, because his like hes daddy,
Paddy got the whooping cough, Felix got the flu,
Paddy died, Felix cried,
he went to the funeral just for the ride,
but he kept on walking still.