Fredericks age in 1940 was 23 and it just says 'Soldier' as his occupation.
May have a possibility, if that age is right.
Best matching birth reg is
March 1917 Chorlton 8c 1266
Wilcock, Frederick
mother Bailey
There's a possible sibling registered in Chorlton in the June quarter 1914, John Thomas Wilcock, mother Bailey.
Going back, using the new GRO birth index, there is another one
WILCOCK, JANE
Mother's Maiden Surname: BAILEY
GRO Reference: 1910 D Quarter in CHORLTON Volume 08C Page 813
So we can look for her in the 1911 census free index
In South Manchester, Chorlton upon Medlock
Jane Wilcock, daughter, apparently of Jonathan and Fanny
https://www.familysearch.org/ark:/61903/1:1:XWBG-J5V
From the above link
Fanny born c 1878 Swindon WiltshireCould this be Fanny Bailey with her parents on 1881 census?
Piece: 2015 Folio: 117 Page Number: 34
Edward Bailey 27, bricklayer
Mary A. Bailey 26
Fanny Bailey 3 Swindon, WiltshireElizabeth A. Bailey 1
Fanny BAILEY
mmn GILES 1878 M Quarter in OF HIGHWORTH Volume 05A Page 18
Elizabeth Ann BAILEY mmn GILES
1879 S Quarter in HIGHWORTH Volume 05A Page 7
Fanny Bailey born
13 Dec 1877Baptism 3 Jul 1878 Stratton St Margaret, Wiltshire
Father:
Edwin, bricklayer Mother:
Annie, abode Gorse Hill
Elizabeth Anna BaileyBaptism 9 Sep 1881 Swindon, St Paul,
Father:
Edward, bricklayer Mother:
Mary Anna also bpt same parents
Edward Bailey 9 Sep 1881 Swindon, St Paul
Fred Bailey 11 Jul 1884 Swindon, St Paul