Hi Cheryl
This is a tough one. I've got some incompletely-formed theories floating around my head so I could be on completely the wrong track with this.
First, though, some baptisms that may be of interest:
Nottingham St Mary baptism01 Dec 1878 - Annie dau of Edwin and Sarah ATKINS, of London Rd (born 20 Sep 1866)

(not sure if this is relevant)
Bulwell St Mary baptisms30 Jun 1886 - Joseph son of Edward & Sarah ATKINS, Collier of 1 Betts Yd
26 Mar 1890 - Robert son of Edward & Sarah ATKINS, Miner
Hucknall Torkard St John baptism06 Jun 1888 - Sarah dau of Edward & Sarah ATKINS, Collier of Orchard Street (she died in 1889)
This point forward could be a complete red herring. There is this intriguing 1881 census record:
1881 census: 12 Lotus St, Nottingham St. Mary, Notts (RG11/3354, f 88, p 28)
Alfred Beals, Head, Mar, 25, Coal Miner, Staffordshire Bedworth
Sarah Beals, Wife, Mar, 23, Staffordshire Tamworth
Lilly Beals, Daur, 5, Scholar, Staffordshire TamworthMatilda Beals, Daur, under 1 mo, Notts Nottingham
John Beals, Son, 13, Scholar, Staffordshire Bedworth
On the 1891 census, if appears Alfred Beal is back in Tamworth living with a new wife Ellen. His 'son' John by then age 22 was also there. The 1871 census seems to show that John was Alfred's brother, not son (Alfred was about 12 at John's birth).
Matilda Beals was born and died in 1881. You can find her birth and death on FreeBMD under Beale. Also, she was buried at Nottingham St. Mary on 30 Jul 1881, age 4 months, abode recorded as Duncan Street. I'm not sure whether Lilly was Alfred's daughter or not - I can't find a birth for her under Beal(e/s). She may have a different father.
So far I have found no certain connection between the Beal and Atkins families, but there seems some curious coincidences between the families. If Sarah was still married to Alfred (assuming she married him, which I'm not confident she did), she would not have been able to lawfully marry Edwin, so perhaps they didn't bother (hence the missing marriage).
One thing that is making me doubt this is that there is no sign of your grandmother, Mary. However, she may have been born in late 1881 or 1882 judging by her age in 1911.
I think it would be a good idea for you to purchase one of the Atkins childrens' birth certificates. This will give Sarah's maiden name and be able to tell us whether I'm completely wrong on the Beal theory.
Regards
Alexander