The pair will work as a creative team across all of BBH's accounts, including Barclays and Levi's.
Allsop and Veksner have spent the past seven years at DDB. Their "cops" ad was the most awarded print ad in the world in 2004, and picked up the Press Grand Prix at Cannes the same year.
The pair's TV reel also includes recent commercials for Volkswagen Eos and Jetta and the launch campaign for the Channel 4 offshoot More4.
The BBH executive creative director, John O'Keeffe, said: "This is a fantastic addition to the BBH creative department. As ever, we want to attract the very best talent and I think Nick and Simon are precisely that."
Allsop and Veksner met at Oxford University. Allsop went on to study at St Martin's College of Art and Design, while Veksner went to Watford to study copywriting.
They first teamed up in 1996 at Banks Hoggins O'Shea and then moved to Ogilvy & Mather in 1998.
Veksner said: "We love DDB, but after seven years, we have achieved everything we wanted to achieve and were looking for a change. Who wouldn't want a job at BBH?"
Earlier this year, BBH hired Mark Reddy from McCann Erickson for the newly creat-ed position of head of art.