That's unfortunate, most inconsiderate of them not to have waited until after civil registration! You are unlikely to get the father's name from the parish record entry unless he happens to be one of the witnesses, and the witnesses can vary from being church officials to family members, so nothing is guaranteed. If you know the date (is it on the IGI?) you may be able to order a copy of the parish register entry.
It would be worth a post on the Lancashire board to ask how and where to order it from.
I wouldn't rule out Warrington for the baptism either, pity they haven't transcribed the dates you are looking for yet.
Jean