I have recently ( 2016) been contacted by
John and Diane Upton who have very kindly sent some more information and two new photos about the company. Alfred Barker(1) is Diane's great great grandfather.
Alfred Barker & Co was established in 1876 at Wrentham Street, Birmingham, their business was 'General Woodturners'. The photo below shows the company in 1911,
second from the left is Alfred Barker(1) and third from the left is William Barker, his son. The boy is Alfred Barker(2)
Below is a photo of Hilda Barker, aged 5 in 1933 riding one of the company's horses. This horse was given to Hilda's frist school, St Alban School,
Birmingham when she joined.
This fascinating photo appeared in the Rocking Horse Magazine in July 2000 ( issue No 4 ) and it is reproduced here
with permission and many thanks to Anthony Dew of the Rocking Horse Shop, Fangfoss, Yorks.
The man with the white hair is Alfred Barker(1). The other two men are sons of William Barker, Fred at the front carving and Alfred(2) bending down to a rocking horse.
John and Diane think this photo was taken shortly after the move to Irving Street ie around 1922. 
Barker formed a partnership with Norton in 1921 and worked from 157 Irving Street, Birmingham. ( Victory Works). Rocking horses, toy horses and many other wooden toys were produced.
But the partners fell out in 1932 and Barker bought Norton's share of the business. The company was then called William Barker as Alfred(1) had retired and they continued making rocking horses
and toys as before.
By 1939 they were fully occupied with war work, the company by then being known as William Barker & Sons Ltd. After the war they made furniture until the company ceased trading in the late 1960s. 
The Barker horses that I have found are very distinctive and easily identified. The following pictures show some of
the important features.