I'm finding it virtually impossible to separate the two William Youngs baptised in Luton in 1777.
There's only one marriage of a William Young in Luton in the right period - to Cornelia Hooker. There's no burial of a William Young in Luton 1777-1823, followed by your William in 1824. There's no William Young in censuses born 1777 Luton. So one or other of the Williams drops off the radar screen
If traditional naming patterns were followed then William and Mary seem to be favourites, as William and Cornelia's first son was named William, and their first daughter Mary (after Cornelia's mother, Mary née Day), the next son Richard after Cornelia's father. They couldn't name their next daughter Mary after William's mother as they already had a Mary - but in fact she died a few months before their second daughter was born so they could have done.
I still think though that William son of William Young and Mary Richardson may have been illegitimate, as there's no such marriage in Beds at the right time, nor any other children as John pointed out. And naming the mother's maiden name is inconsistent with the practice in the parish register at that time. Not that possible illegitimacy has any bearing at all on whether or not he was the William who married Cornelia. But I've also checked against William Robinson as if he were illegitimate he might have been known by his mother's name (which was the norm) but again I can find nothing.
The best place to start looking is the parish register and the 1803 marriage. What other detail is included in the marriage entry? Is a parish of residence other than Luton given for William? Who were the witnesses? Unfortunately the transcripts from which we usually work don't include witnesses, so you have to refer to the microfilm, which you can do at your local LDS Family History Centre. Alternatively BLARS would look it up for you, for which a nominal charge would be made.
David