Presumably from previous replies, the marriage record tells you nothing.
Your only option is to get the baptism film and have a look through until you find William's birth. Just because there is no record on the IGI it doesn't mean that it's not on the film. I've found lots of information which the LDS haven't transcribed. Bulkington films are at Bedworth library, Coventry family history centre, Warwick records office and Lichfield records office, or you could order a film from the LDS to view at your nearest LDS centre. But no, they are not online, very little is for the dates you are looking at.
Finding the birth may not give you very much more information - it's likely to just say William son of (say) William and Mary. However Bulkington records do go back to 1660 so it would be well worth spending a decent amount of time looking at the film and recording all instances of the names you are looking for and then see what sort of pattern you can build. Information is limited at this time so there is never any guarantee that you have the right people and it's the sort of research you really have to do for yourself as it would be unfair to ask someone to look it all up for you unless you paid a professional researcher.
Also think about checking the records for nearby churches. I have found Bulkington people using Shilton, Ansty, Wolvey, Bedworth, Nuneaton, Chilvers Coton, Foleshill, and Walsgrave on Sowe.
You could also look at other sources such as apprenticeship, poor law and newspaper records to see if you can spot him. These often give clues to parents. The poor law index is available at
http://www.midlandshistoricaldata.org (pay per view)and there are Dewis's listed although I haven't checked the dates.
Hope your search is fruitful.
Kate
