There are 4 Ernest Smith births in Nottingham in 1913, but without even being sure that that's where he was born, it will be a search for a Smith in a haystack of Smiths.
There are 3 deaths for Ernest Smith in Nottingham in 1993 (this assumes that the ancestry tree is correct), with yob between 1909 - 1913 (all of which are subject to informant error):
SMITH, ERNEST 1913
GRO Reference: DOR Q1/1993 in NOTTINGHAM (6891E) Reg E52A Entry Number 194
SMITH, ERNEST 1912
GRO Reference: DOR Q1/1993 in NOTTINGHAM (6891D) Reg D52A Entry Number 116
SMITH, ERNEST 1909
GRO Reference: DOR Q1/1993 in NOTTINGHAM (6891A) Reg A54A Entry Number 152
I think the best way forwards is for the OP to read this thread, and see if one of those Smith/Carding births might be her father. If that looks possible, then get the 1937 marriage cert to find Ernest's father's name. From there it might not be too difficult to track down Ernest's siblings.
(And remember, Edie might be a shortened version of Edith, a petname, or a middle name ---)