Just to return to this for the interest of everyone who helped...
Having spent far too many hours on it, I still only have a likely scenario, but obtained a few more hints from various records from Bucks FHS and also found volumes of transcribed of Beds. parish records, Cranfield here:
https://archive.org/details/bedfordshirepari27bedfWilliam Byway was possibly born 1755 in Bow Brickhill or North Crawley - the only possible records in Bucks. And I think he probably did first marry Mary Barfoot in SG in 1790 and have daughter Elizabeth in 1791. I got all excited when I saw on an FindMyPast header record that a Mary Byway was buried in Cranfield in 1797 (which would have led nicely to William marrying Alice Odell in August of the same year), but Bedford parish transcript above shows a Mary w. of Thomas Byway (labourer) dying in that year, the same year as their son dies, so that can't be Mary Barfoot. :-(
However, the William who marries Alice in Cranfield is a widower, so there's a good chance it's the same one who married Mary B, lacking any other records contradicting this. I've traced Alice Odell's parents, and also found that Alice Byway remarries in SG after William's death, to a William Parkins, and there's a Thomas Chapman who is a witness at William's marriage to Mary Barfoot as well as Alice's marriage to Wm Parkins.
So unless I come across any contradictory evidence, I am going to stick to this story. It's a shame I can't find a definite death record for first wife Mary Byway, and it would be interesting to find how the Thomas and Mary Byway couple in Cranfield are connected to the Bucks lot, but this is as far as I'm going with this one.