If you order one of the children's birth certificates that should give the mother's maiden name. Is it possible she was a second wife? I would start with Lydia, as she is most likely to have been Mary's child.
Who was the visitor Mary Neal born Ireland in 1871? She may be a relation.
Births Sep 1858
Wingate Thomas Bromley 2a 216
Births Sep 1860
WINGATE George Bromley 2a 235
Births Jun 1862
WINGATE Mary Bromley 2a 259
Births Jun 1864
WINGATE John Bromley 2a 271
Births Mar 1868
Wingate Elizabeth Bromley 2a 346
Births Jun 1870
Wingate Lydia Susan Bromley 2a 339
Ordering through the GRO is the cheapest way.http://www.gro.gov.uk/gro/content/certificates/