How likely do you think it would be for two sisters baptizing their children on exactly the same date and both name their daughters Annie & Anne
I've had a look at the actual parish records on the Lincs to the past web site and it has both christenings for Annie Smith daughter of Charlotte and one for daughter of Emily.
Maybe Emily and Charlotte were sister's or cousins.
It may be why the GRO was unable to send you the birth certificate because there was 2 Annie's born and didn't know which one was the right one for you.
You mentioned previously you thought that
Annie Smith age 9 Claypole on 1881 census (
with parents Francis & Ann) perhaps should have been noted as their Granddaughter
Could it be that this Annie Smith on 1881c is the daughter of Emily Smith and it’s got confusing as she has a sister named Charlotte bn c1851
Looked at all Francis & Ann's children from 1851 – 1881 census and their birth entries on GRO indexes confirm that their
mmn ALDRIDGE there is
NO entry on GRO indexes for an Anne/Annie Smith 1871/72 with mother’s maiden name Aldridge that I can spot, so your theory of her being a grandaughter of Francis & Ann looks feasible
Their daughter Charlotte Smith appears to have married in Sheffield 1872 to a Joseph Corah (as previously mentioned m/c would confirm father's name)
and she can be found with her husband on 1881 census living next door Francis & Ann Smith (most likely her parents) and of course Annie Smith age 9 Claypole
Charlotte & Jospeh Corah had one son Thomas
CORAH, THOMAS
mmn SMITH 1885 S Quarter in NEWARK Volume 07B Page 427
Thomas Corah died age 4 Dec 1889 Newark
and it looks as if his mother Charlotte possibly died during or soon after childbirth
Charlotte Corah Age at Death: 34
Jul-Aug-Sep 1885 Newark Volume: 7b Page: 214
Haven't done an extensive search for Francis & Ann's dau Emily
This might be her on 1871 census
RG10; Piece: 3542; Folio: 57; Page: 24
Robert B Tresham 39
Sara Tresham 29
Percy Robt Tresham 5
Florence Tresham 3
Elizabeth Mary Tresham 2
Emily Smith 18 Newark servant
Mary Gibson 50