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
Jonathan married previously here he is with dau Alice on 1901 census, wife is Elizabeth
Piece:   3694 Folio:   88 Page Number:   10
62 Rusholme Rd    
Jonathan Wilcock   36 Blackburn , Paint Decorator House Elizabeth Wilcock   40 Northumberland Restaurant Proprietors Coffee
Sarah J ECCLESTON dau   21
Mary C Eccleston dau   18
Robert H Eccleston son    14
Alice Wilcock dau   7 BlackpoolCharles Files boarder    43
Alice Atchin serv   34
Sarah Atchin    3
and here's his marriage to Elizabeth
Jonathan WILCOCK age 36,
 painter, 2 Stand Street
Father Jonathan, Farmer
Elizabeth ECCLESTON 39, widow, 2 Strand Street
father Edward PRINGLE, Inspector of Police
married on 19 Sep 1900 at St Andrew Ancoats Lancashire
Witnesses Frederick Townsend & Mary Catherine Eccleston
1891
Piece: 3401 Folio: 143 page 44
Jonathan Wilcock   63 Hide dealer
Margaret Wilcock   57
Jonathan Wilcock   26 Blackburn Lancashire cab proprietor
1881 Piece:   4189 Folio:   8 Page 9
Jonathan age 16 Blackburn with his parents  occ painter