I think that "the registers have not been deposited" means they are still at the church. This is a serious drawback ... no baptisms, no marriages (him or his mother), no burials, no nuthin. I was going to mention that I could possibly have a look in the Archives next week, but there's no point as the register is not there!
He was possibly born on St Swithin's day (15 July), his father may have been a CASH. It seems you/your friend has a choice ...
1) Marriage cert - perhaps a father's name, perhaps not, or something made up

2) Birth cert - a mother's name
3) Find out what can be gleaned from the church at Burwell - if you can find who to contact

Perhaps seek advice from here (they should know how to contact Burwell)
http://www.stjameschurchlouth.com/context/parish.html