I've done a bit of Googling and as Paddyann says, if the baby lived even a brief time, there should be registration certificates.
If he was sadly stillborn, there is something called the Stillbirth Register. Sensitive link. www.gov.uk/register-stillbirth
Apparently copies of the latter certificates are only issued to the parents of the child concerned, or, if the parents are deceased, any siblings.
Phoning the likely registry offices might be productive.
Parents-in-Law. What do/did you call them?