Hi, I have found a William Godwin specifically born in Bristol, however there are others in Gloucestershire around this time as well in case this is not him, as it is a year later than you said (but leeway can be given) I found the 1901 census in Waterford for William, do you have this also?
1881 census from findmypast.co.uk (living in Barton Regis)
Edward Godwin Head Married Male 37 1844 Carrier Bristol, Gloucestershire, England
Transcription
Sarah Godwin Wife Married Female 36 1845 - Dorset, England
Transcription
Elizabeth Godwin Daughter Single Female 13 1868 - Bristol, Gloucestershire, England
Transcription
Edward Godwin Son Single Male 8 1873 - Bristol, Gloucestershire, England
Transcription
William Godwin Son Single Male 3 1878 - Bristol, Gloucestershire, England
1891 census
Sarah D Godwin Head Married Female 40 1851 Tailoress Whitechapel, Dorset, England
Transcription
Edward J Godwin Son Single Male 18 1873 Hair Dresser Assistant Bristol, Gloucestershire, England
Transcription
William Godwin Son Single Male 13 1878 Painter Bristol, Gloucestershire, England
Transcription
Wm D Williams Boarder Single Male 23 1868 Baker Blaenavon, Monmouthshire, Wales
I think this is most probably your William's family as he is also a hairdresser on the 1901 census and his brother is a hairdresser here. I'm interested seeing as I have Godwin family in Gloucestershire and Wiltshire
I think this is his parents' marriage
Edward Godwin to Sarah Denning Gapper 26 May 1867 St Jude Bristol. Parents John Godwin and Benjamin Tapper.
Christenings of Edward/Sarah children:
Elizabeth Ann 1868
Edward John 1872
In Bristol.
Let me know if you need any more help, or if you think this is the right/wrong William. Do you have the marriage certificate of William, as this will confirm who his father is