Hello Dave,
Sorry, I am too far away from Chester to go myself.
I think that, in some cases, there can be records in the Quarter Sessions papers regarding illegitimate children.
Have you looked for his marriage? The marriage certificate might show his father's name. It is possible that the entry in the Parish Registers would include this if you know where and when he married.
I say this as my gt gt grandfather was illegitimate and his marriage certificate had his father's name on it.
The Johnson family were quite extensive in that area of Cheshire in those days. I have some in my tree.
Jo.