Stephenson Harwood originally won the contract in 1999 after a tender process. It is understood that Field Fisher Waterhouse and Nabarro Nathanson also bid for the work.
But in August, Stephenson Harwood trademark partners Tibor Gold and Nick Hedley announced they were quitting for patent agency Kilburn & Strode.
Stephenson Harwood decided not to replace them and to close its intellectual property (IP) practice.
Regardless of the firm's actions, the departure of Gold and Hedley meant the BBC would terminate the contract. Under the terms and conditions, it retained the right to end the agreement should either partner quit. The clause also included assistant Mary Blahene, who has returned to Ireland to take up a post with A&L Goodbody.
The BBC retendered the work and Bristows emerged as the winner. It is not known whether Field Fisher or Nabarros re-tendered, although Gold's new firm Kilburn & Strode did compete.
The BBC work involves filling and obtaining trademarks, global recognition, defending trademarks against attack - often from "cybersquatters" - and occasional prosecution after abuse.
The BBC handles litigation in-house but the trademark contract does include "with notice" work.
For the financial year ending 31 March, the BBC spent £1.6bn on programmes, many of which were sold around the world. Its commercial arm, BBC Worldwide, saw a 15 per cent increase in turnover to £82m for the same period.
It is understood that Stephenson Harwood and Bristows are going through handover period. Bristows will take over the work full-time in the New Year, with trademark attorney Madeleine Clarke handling it.
Elizabeth Rouse, the BBC's most senior IP lawyer, declined to comment. The IP department has been without a head since December 1999, and the position is under review as part of the broadcaster's on-going strategic reorganisation.