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