Baker & Co were a ‘West Country’ firm founded by Charles Baker in 1848 who opened their first London premises at Seven Sisters in 1864 as ‘Chas. Baker & Co.’ They expanded steadily across the city with outlets in Fleet Street, Oxford Street, Tottenham Court Road (plus a few others) and a main headquarters at 271-272 High Holborn. Note that their early London advertisements make reference to “Real West of England Clothing”.
Initially they traded as ‘Chas. Baker & Co.’ until Charles Baker established the company as ‘Baker & Co.’ in 1877, with Joseph S. Baker being admitted as a partner in 1878, and they became an incorporated (‘Ltd’) company in 1899. In 1914 the company directors included G. Edgar Baker and Clifford P. Baker along with Joseph S. Baker (the latter as Chairman and Managing Director).
They weren’t button manufacturers though and your button would have been made
for them rather than
by them. They were specialists in gentlemen’s clothing, school outfits, and also made uniforms for some branches of the Royal Navy, colonial parts of the Navy, the Merchant Navy, and commercial shipping lines. They also had retail outlets linked to “The Sailors Home”, a charitable institution operating in a number of British ports (including London) that looked after the interests of those in maritime professions.
They seem to have disappeared from London sometime around the start of the war in 1939 although it isn’t clear if they were still operating elsewhere or whether this absence from London records was temporary. I have seen uniforms for the Royal Navy Reserve in ‘Pattern 1958’ that still bear their name. Certainly they were still in business (or had resumed operation after the war ended) as ‘J. Baker & Co. Ltd’ Royal Navy Outfitters after the war, as per this 1947 advertisement.
Note however that there is no London address given at this time. Portland, Portsmouth and Devonport are all on our SW coast.