Hi there,

If you search for a Constance Freeman born 1897 in Radford she appears in 1901 census living with the Hallams. So it would appear that Constance Annie probably was her birth name.

With regard to a birth record this is a possibility but it is a year out, considering the circumstances I would say there is a good chance it is her:

Births Dec 1896

Freeman Constance Annie O Nottingham 7b 306

Hope this helps
Amelia