Being born so close to the end of June his birth will more than likely be registered in the 3rd quater, July-Aug-Sept. There is a William Taylor registered in that quater born Whitechapel, details below, would be worth ordering the birth certificate to find out for certain.
Name: TAYLOR, William
Registration district: Whitechapel
Year of registration: 1878
Quarter of registration: Jul-Aug-Sep
Volume no: 1C