According to <<https://my.strathspey.org/dd/issue/2644/|issue 2644>> Longman & Broderip were active in London in the second half of 18th century as publishers and musical instrument sellers. They were based in Cheapside and Haymarket until going bankrupt in 1798.

Further details from <<https://my.strathspey.org/dd/issue/3300/|issue 3300>>:
Their full names were James Longman and Francis Broderip.  
This English music publishing house, music retail shop and instrument seller was founded in London in 1767 by James Longman.
In 1769, Lukey joined the firm and in 1775, Francis Broderip became shareholder of the company.
In 1798 the company became insolvent and the former owners went on separately.