To find this company one needs a Kelly's directory for Warwickshire from the late 1930s or 1940s.
One has to find an incorporated company with the proper tools to make buttons active in a non military trade. During wartime all governments distributed materials and handed out orders to manufacturers.
E.g. the leading USA compact maker manufactured US Army buttons during WWII. There must have been a shortage of compacts during WWII.
I could not find a downloadable Kelly's from the period and I have tried my luck on a 1914 Trade Directory.
Note: these are only options! No solution given!
Anstey and Wilson were a Birmingham-based electro plated ware manufacturing company and appear to have been trading right up until the late 1990s when it was dissolved.
also stainless steel dinner knives
Anstey & Wilson limited. Incorporated 13 March 1957, dissolved.
Not a candidate if there was not an earlier incorporation.
Alabaster & Wilson, 11 Legge Lane, goldsmiths, jewellers manufacturers
Alabaster & Wilson Ltd. established 1887, incorporated 1947.
http://www.alabasterandwilson.com/
I am expecting wartime production. As they were incorporated in 1947 they are not a candidate.
Altendorf & Wright, Vesey str. St. Mary's - gun stock makers and dealers, tin plate workers.
A strong candidate, but dissolved in the early 1930s.
A valid suggestion made by member RSM:
Charles Herbert Atkins & Ernest Wigley, 171 Hockley Hill, Birmingham (c.1920)
not in 1914 trade directory
1914: Wigley & Hall, 87 Vyse St. as jewellery factors & wholesale jewellers
1914: F.W. Atkins, 106 Vyse St. Jewellers manufacturing
no trace of being incorporated.
not researched:
Alexander & Ward, 19 Claybrook St., music & musical instrument makers & dealers, piano tuners.