Pleasley was in Mansfield registration district.
Looking at the period from 1921 to 1925, this seems most likely to be the death of the neighbour (in fact it's the only death of a mother-aged woman in that district during that period):
Sarah Bingham, aged 36, June quarter 1922 (volume 7b, page 137)
So she was born in 1886 (had 36th birthday in 1922) or 1885 (would have been 37 in 1922).
There are three possible marriages in Mansfield:
Watson Bingham and Sarah Woodward 1905 (definitely married)
William John Bingham and Sarah Allsop 1906 (may have married)
Richard John Bingham and Sarah Ann Lambert 1908 (may have married)
Births after 1911 show the mother's birth surname in the index.
For Bingham-Woodward, there are numerous births, the last being a boy in September quarter 1921.
No Bingham-Allsop. And no Bingham-Lambert.
So it seems quite likely that the Bingham-Woodward couple were the neighbours.
It would be a little cumbersome to list all the children here; just go to
http://freebmd.rootsweb.com/cgi/search.pland search for Bingham births with mother Woodward and select Mansfield district; no other info needed.
It's also wise then to check whether any of the children died in early childhood.
Some of the names are fairly distinctive so you could then also search for their marriages.
The couple had other children, born before 1911: Ada, Sally, Edith and Caroline (who probably died, as there is a later Carolyn -- or no, may have married in Nottingham in 1931 and died in 1996 in Berkshire). They are living with Watson's father in 1911 in Pleasley; he was a miner hewer. (Looking at a family tree, see below, it seems that the eldest children were from Watson's first family; he was widowed very young.)
Watson Bingham born abt 1878 died in 1951 in Mansfield.
If you go to
www.mundia.com and register (free of charge) you can search family trees for the names of the children whose families you would like to contact and send messages through the system; the tree owners receive them through Ancestry, where the trees at Mundia are taken from. One of the younger children was also named Watson, a distinctive name to search for, and he is in a tree that lists all the children and includes living family members (whose details are not shown, for privacy). Actually it seems to indicate that some of the children are living, which seems unlikely, but it does have death dates for several of them, who lived to the 1990s. It may just be that the tree owner doesn't have death details for the rest.
The tree shows Sarah as having died in 1922.
That tree owner seems to know a good deal about the family, so this could be a very easy way to make contact with them.
There is no obvious Bingham-North connection here yet but that would mean searching back (and knowing names and dates in your wife's mother's family) -- I wonder whether your wife's mother was best friends with one of the Bingham girls about her age, Caroline born in 1910 or Lucy born late 1911 or early 1912, for example, and the photo was a keepsake when she emigrated. Lucy's info is private in that tree, and I'm not having much luck tracing a marriage or death for her.
I hope that helps!