Hi Daryl.
People did not always know where they were born, often they would give their birth place as the earliest place they remembered living, with this in mind I suggest you take a look at the family living in 1871 in Crispin Street, St Sepulchre, Northampton RG10 piece 1483 folio 97 page 3
When entering Crispin Street and Kingsthorpe into a map the distance is shown as half a mile, suggesting that this is your family.
They have a son Alfred J aged five born in Bermondey, there is a birth of Alfred John WRIGHT in 1865, Sept. quarter, Bermondsey, vol 1d, page 90 which is likely to be your Alfred who you say was born about June 1865. You have six weeks in which register a birth, so it would not be unusual for someone born in late May or in June to appear in the September quarter.
Jebber
The offical GRO indexes now give the mothers maiden name on all births.
So it looks like Alfred's mum's maiden name was Clay.
WRIGHT, ALFRED JOHN mmn CLAY
GRO Reference: 1865 Sept Quarter in SAINT MARY MAGDALEN BERMONDSEY SURREY
Volume 1D Page 90
And his parents marriage...
Marriages Jun 1861
Clay Mary Hannah Elizabeth in Newington 1d 210
Wright Charles in Newington 1d 210