We all have only a little to go on when starting out!
To summarise my understanding of what you have:
Norman J Cooke was brought up in North London with probably 2 brothers, William & Thomas.
Their parents were William Charles Cooke & Annie Jane Thomas. I'm guessing you know this mainly from documents found when Norman Cooke died, since you say it was then that you found out Norman was born in the Penzance area. (Birth can be seen on FBMD September Q 1916.)
Norman's parents moved to Bishops Stortford in Hertfordshire in the 1950s: their deaths can be found on
www.freebmd.org.uk in the Bishops Stortford RD:
William C Cooke March Q 1954 age 66 – so birth c1888
Annie J Cooke September Q 1967 age 82 – so birth c1885
PollyQ has found you a marriage between a William C Cooke & an Annie J Thomas in the Cardiff RD March Q 1914.
Now that you say Norman probably had 2 brothers Thomas & William, & bearing in mind the locations we now seem to have of Cardiiff, Penzance & North London, I have just searched FBMD for births to surname Cooke with mother's maiden name Thomas between 1914 and 1934:
That finds me just the one Thomas nationally : Thomas S Cooke in the March Q 1915 Cardiff RD.
There are 3 Williams, all in 1922, born Edmonton, Leigh & Monmouth. I'd be thinking the chap born in Edmonton would probably be of interest to you, what with Tottenham being in that RD, and William's middle initial being “C”....
Then on
familysearch.org/we have indexing of WW1 service records 1915 Cardiff then 1919 London, which look promising; I believe the detailed service records are viewable on either ancestry.co.uk/ or findmypast.co.uk/, which are pay websites. If you are in the UK, one can usually access Ancestry at your local public library (with library card); alternatively both websites often have a short window of a few days around 11 November when everyone can access armed service records from home.
As you know Norman's father William C Cooke was a hairdresser, these service records could be helpful in mentioning occupation – & also Next of Kin.
Then I would also suggest you get to view the 1911 full census entry (either Ancestry or FindmyPast) where there is a possible entry of interest Wm Cooke born Penzance age 23 in Tredegar S Wales.... that will give you an occupation as a minimum.
I may be mistaken in considering the musician Cooke family in Penzance 1891 census
www.freecen.org.uk/ as a feasible candidate to be the family you're after, but I note that, in addition to there being just the one birth nationally for a William Charles Cooke that is an exact match for Norman's father per his recorded death age i.e. birth Penzance RD December Q 1887 which looks very much like the young William in this family, that William C Cooke's younger brother is one Thomas Sylvester Cooke.... now what was the name of that Cooke child born in Cardiff a year after the 1914 marriage PollyQ found?
??
The remains of this family are still in Penzance 1901.
Oooh, this is getting interesting: I can see the eldest son in this musician family, in 1901 enumerated where he proves his mother's maiden name..... then later crossing the Atlantic.... but you will want to find some of this for yourself, and besides, you need to firm up the identity of Wm C Cooke born c1888 first.......
(And P.S. I think I can see the 2 later life events for Thomas S Cooke 1915, putative brother of your William C Cooke, in the Edmonton RD: married 1942, died 6 months later age 28. Again, this is a possibility only, but circumstantially promising.)
But we still haven't discovered how you know Norman's mother Annie Jane Thomas was born St Just...........?
Sue
?? 