The only Shaw I have info on is my Great Grandfather, Miles Shaw. He was born on May 8th, 1886 in York Co, Nebraska. He filed for his social security card in Waco, NE. His father is listed as John Shaw and mother was Hannah Denney. Miles Shaw died on 3/29/1965 in Fresno, CA. I don't know if Miles Shaw had any brothers.
My grandfather did have 2 sons. One son was named Howard Shaw, born 1926 and another son named Rodger Shaw, but I'm not sure when he was born. I do know they were both born in York, NE.
I don't know if any of this info would be helpful to you. As you can gather, I am looking for info on a John Shaw who was probably born in mid 1800's.