I've been skimming through Lincolnshire parish records today, accessed through 'Lincs to the Past'. In looking through a parish burial register I found the following entry for an infant girl. After the fact (i.e. after the burial) the parish priest added a terse note:
Buried with Christan Burial inadvertently ; the father having falsely stated that the child had been baptised.
This also accounts for the statement of her name: a child called Sarah Barsby - till baptised she did not have that name in the eyes of the Church.
If it affected whether my daughter was given a funeral, I'd lie too.