"Both those children registered as illegitimate with no father mentioned (mother Rachel not informant for Lizzie's birth but she was for Samuel's). It could still be that William was the father but she didn't want him on the certificate. Another possibility is that one of William's brothers was the father of one or both children. Equally possible that William and Rachel were living apart and William was the father but informant didn't know 1) who the father was and 2) that they were married."
Lizzie is my granny. DNA doesn't lie, William is the father. I'm not sure why he was not listed. He is listed on John Walsh birth certificate as the father
He was quite a character going by all the records and newspapers articles I found out about him. He had 2 other wives after Rachel, he was convicted of bigamy for both of these. In the second bigamist marriage his sister Mary and her husband Thomas Dougall were witness.
He also had children by a third woman Mary Mullen. The first child William was born in 1923. On the birth certificate Mary is listed as Mary Walsh but I've yet to find a wedding certificate.