A fair amount of information can be found about this in Neil Dickson's "The History of the Open Brethren in Scotland 1838-1999". Here are just a few of the most relevant quotes, but much more can be found by searching for 'Pickering' withn it.

"In 1876 Donald Ross opened the Publishing Office in Sauchiehall Street, Glasgow, and when Henry Pickering (its manager from 1886) formed a partnership in 1893 with printer William Inglis (died 1906), it combined both men's names in its title." p182

"In addition [J.R.] Caldwell may have had a crucial role in the development of Pickering & Inglis for in 1886 The Publishing Office shared premises with Caldwell's firm (Dempster, 'Brethren publishing', p. 70) and JB Watson stated that it was Caldwell who invited Pickering to take up the appointment of bookshop manager (Watson, 'Pickering'). These evidences of control suggest that Caldwell may have also supported the publishers financially during this period. " p183

"It was the restricted views which were winning over significant individuals. Ritchie issued Banks' pamphlet, as did Henry Pickering, then
running a small printer's business in Newcastle. Hopkins' pamphlet was issued by the Publishing Office in Glasgow and Banks, took over
control of this concern in 1886; it was he who appointed Pickering as his manager. JR Caldwell, who was Boswell's brother-in-law, was also won over to these views .. " p203

"In 1919 Pickering & Inglis absorbed Brethren publishers RL Allan of Glasgow, Alfred Holness of London, and Yapp and Hawkins of London. It became a limited company in 1937: 'John Hawthorn', W, 67 (1937), p. 47."  p196

"In 1908 on the death of William Inglis, he (John Gray) became a partner in Pickering & Inglis and he removed to Glasgow, taking the main responsibility for printing and publishing after Henry Pickering retired to London in 1922" p354

From https://eddieolliffe.wordpress.com/tag/pickering-and-inglis/ -

"In 1906, the firm expanded by printing in Glasgow and later Cardiff and opening shops in Glasgow, London, Edinburgh, Newcastle, Dublin, Bournemouth (Keith Jones is ex-P&I), Manchester and maybe some others.  They were a chain before such were known but tended to operate independently. That was their inherent weakness. 

By the 1930s they, along with Marshall Morgan & Scott, were the two leading UK independent publishing businesses. The bookshops were a good outlet for P & I books and their printing output gave them an advantage over MM&S. 

When the new centrally-run, charity-based SU and CLC shops came along, P & I found it difficult to compete and the shops closed one by one. The last to close were Manchester in 1966 and London in 1985. However the large Glasgow shop continued flourish and survived a company merger with competitors MM&S in 1981.

The Glasgow shop was bought by STL in 1999 and became Wesley Owen’s flagship store for ten years before being bought by Koorong in 2009."

In 1981 P&I merged with Marshall Morgan and Scott (a London-based predominantly Baptist publishing house, which had acquired a number of publishing companies over the years, such as Bagsters and Oliphants) to become Marshall Pickering. Marshall Pickering was itself acquired by Zondervan in 1983 and merged into it in 2001.