Hi Greg
As Debra says, two possibilities on board the Tottenham.
William Wells/Welch shows him disembarking from the Tottenham on 20th October 1818 and being forwarded to Liverpool for general distribution. He was one of 97convicts shown in this particular record and the other William Welch is not on that record, but he may be on another one for the same ship.
A Conditional Pardon was issued for a William Welch on 31st Day of August 1837. Description is Ship - Tottenham, Native place Hertfordshire, Trade - Gardener, Offence - Housebreaking, Sentence - Life, Year of birth 1793, Height - 5 feet 8 inches and three quarters, Hair - Black, Eyes - Dark
Noells