Searching local papers via the BNA site enables this potted history of the firm.

Regular adverts thro the early 1890s show the Company had been in existence since at least 1891 as Teague & Chew, Engineers, Cinderford. Seem busy as looking urgently for such skills as pattern makers(ie foundry), clerks, etc.

Making large steam winding engines for mines
http://www.gracesguide.co.uk/Teague_and_Chew
https://en.wikipedia.org/wiki/Parkfield_Colliery

Occasionally contracted by the Council to supply equipment, such as in 1893 for a pump over a well on Ruardean Hill.

Messrs Teague & Chew of Heywood Engine Works, Cinderford, have taken into partnership Mr D. Fleming of Glasgow, firm henceforth will be Teague, Chew & Fleming. The firm has obtained an important order from the Government for construction of dock gates, required in connection with the extensive operations they are carrying out on the Galway coast.
Gloster Journal, March 1896

In 1897, for example, Teague Chew & Fleming exhibited a 1 hp horizontal steam engine & boiler, fitted with Teague's automatic expansion valve, at Gloucester Agricultural Show; alongside such famous firms as the Gloster Carriage & Wagon works, Fielding & Platt of Gloster, RA Listers of Dursley & Gloster, etc.

January 1904 the partnership dissolves, "as Mr Fleming retires", the firm is Teague and Chew again.

In 1907 Westbury Board of Guardians, needing to replace a worn-out donkey pump, after discussing quotes from various makers, bought a £15 pump from Teague & Chew which could throw 400 gallons an hour.

NEW DEAN FOREST COMPANY.
Teague and Chew, Ltd., has just been registered with a capital of £7,000 in £1 shares (2,000 pref.), to acquire the business carried at the Heywood Engine Works, Cinderford, Gloucester, as Teague, Chew, and Morman, to adopt an agreement between M. E. Teague, A. S. Chew, and T. R. Morman of the first part, the said M. E. Teague of the second part, and Teague, Chew, and Morman, Ltd., of the third part, and to carry on the business of iron and brass founders, mechanical engineers, manufacturers of all kinds of implements and machinery, metal workers, tool makers, iron and steel converters, etc. The subscribers are M. E. Teague, Bowson House, Cinderford, Glos., engineer; A. S. Chew. Mitcheldean, Glos., engineer; C. Hayward, Steam Mills, Cinderford, Glos., engineer; W. E. Ruoh, Belle Vue-road. Cinderford, Glos., accountant; Mrs. E. Teague, Bowson House, Cinderford, Glos.; Mrs. A. Chew, Mitcheldean, Glos.; and J. S. Dudbridge, Stroud, Glos., incorporated accountant. No initial public issue. The number of directors is not to be less than two nor more than five : the first are M. E. Teague and A. S. Chew (both permanent special qualification £750). Qualification of ordinary directors £lOO. Remuneration as fixed by the company.
Glos Echo, March 1908

Throughout the 20s & 30s the papers carry adverts for foundry workers etc, suggesting steady business.

FOREST INDUSTRIES
A LARGE PERMANENT CONTRACT
A definite step towards return to industrial prosperity lias been made Forest of Dean engineering firm, who have secured a large permanent contract to supply high-class machine castings. Twenty men have been added to the pay roll, and there is a prospect of considerable expansion, which will mean work for still more men. Two moulder improvers are wanted immediately, and they cannot be found in the Forest or in Gloucester. The firm concerned is Teague and Chew, Ltd., of Heywood Engineering Works, Steam Mills, Cinderford, and their success is triumph of adaptation to the changing trend of industrial needs. At one time the main output was castings for steam engines; the new contract concerns electric machines.
The history of the works was told to "The Citizen" reporter by Mr. C. E. Teague and Mr. B. P. Chew, who followed their fathers in the business. It was founded in 1888 by Mr. M. K. Teague, who was joined by Mr. A. S. Chew. The Forest of Dean was flourishing in those days, and the works were kept busy with the manufacture of Teague’s expansion gear, of which Mr. M. E. Teague was the patentee, and compounding steam engines.
The collieries and iron mines were the principal customers then. There were 65 men employed at the works, and slack time was practically unknown. During the Great War Messrs. Teague and Chew resisted the tempting munitions contracts in order to cater for the local collieries, thinking this would react in their favour later. After the War came amalgamations, and big colliery concerns had their own engineering works.
About this time, too, there was change in personnel, from one generation to another. Mr. A. B. Shew died in 1918, and his son, Mr. B. Chew, returned to the business after gaining experience on the Clyde and at Cardiff.
Mr. C. E. Teague was serving in Palestine with Gloucestershire Yeomanry from 1914 to 1919. He won tho D.C.M and was twice mentioned in despatches. Mr. M. E. Teague died in 1924, and now the sons are following in the footsteps of their father, and their heritage gives them something stronger than an ordinary business relationship.
Since the steam engine has receded into tho background, Messrs. Teague and Chew have extended their activities to general engineering, and they have manufactured a considerable amount of tinplate machinery. Even the advent of the electric machine has not found them wanting, and their new contract has brought them to the forefront of modern industrial enterprise, after sharing the slump of the last lew vears.
"This new work is keeping tho iron foundry very busy" Mr Chew said, but we have plenty of scope to deal with further orders, especially in the machine shop."
Mr. Teague said: "This contract permanent and can expect that others will come our wav from the same source. We will expand our works as the need grows, but that, of course, will be gradual. It is certain that we will want more men before long.”
Gloster Journal, August 1934

Reading about just this one firm from my small hometown gives great pride & some sadness to this mechanical engineer. I had to leave the Dean in the 1980s to find work, and now most of the companies I worked at are just memories, even large and famous aerospace firms are a shadow of what they were just 30 years ago...
Thanks so much to this website for helping me discover what a busy workshop the Forest once was...