Karen, you say you have a Bastardy Warrant. I am familiar with Bastardy Bonds but have never seen a Warrant. Would you mind sending a copy to me please at my email below.
If you look at The Bradfield Arcives site there are many records for Mary Walker, though it was a common name so maybe none of them is yours. There is also one record for a Samuel Jeffcock, a grinder.
Do you have your ancestor in any of the earlier censuses to be able to see if he had any siblings?
Regards, Dave