Hi Doreen,
Unfortunately, although we have a lot of information about William Buer b.1749, we have not found his birth record. We identified his father from an aprentice record from the Merchant Taylors - which notes his father as Joseph Buer from Parndon, Essex. The Essex on-line records for Parndon do not quite go back to 1749, so perhaps a trip to the actual archive would help.
I have just realised that you may be refering to William Buer b.1813 (Wm Buer b.1749 is your William's Great Uncle). What we have for your William is that he was the son of Charles Buer's second wife Judith Stone and he was born in Middlesex and baptised/born 7th Nov. 1813. I think my sister may have a copy of the actual record (I will try to get the a copy and post).
Do you have any information, documents or pictures for this branch of the Buers?
Regards
Max