Your boyfriend should phone the UK Passport Service on 0870 521 0410 for advice. They will try to provide him with an appointment at the Passport Office which serves your region. (There's normally a 10 day wait for an appointment but UKPS usually keep a few appointment times spare for genuinely urgent cases).
UKPS will require a signed statement relating to the circumstances of the loss. As it seems unlikely that the child's mother would be willing to provide such a document, I suggest asking your boyfriend's solicitor to provide a signed statement that he has been advised, by the mother, that the passport has been lost.
I also suggest advising the girl's mother that you intend to follow this course of action. Point out that the issue of a new passport will invalidate the 'lost' one. This alone might result in the missing passport suddenly coming to light.
These links might help:
http://www.passport.gov.uk/general_preventing_ law.asp
http://www.passport.gov.uk/passport_replacing. asp
Chris