French joined Shell in 2011 as a senior spokesman, having previously worked as a freelance PR consultant and Conservative Party aide.
He will be taking over from Megarry, who has moved to Saudi Aramco, the giant Saudi Arabian oil company, as a senior communications consultant.
French, who will be running a team of five, said: "It's a great privilege to lead a fantastic team of communications professionals at one of the world's largest, and best, companies. This is a great opportunity to develop my career in a truly fascinating industry, and I relish the challenges that lie ahead."
Andy Norman, Shell's vice-president of media relations, said: "Jon's promotion is deserved recognition for his consistent delivery, calmness under pressure and well-developed leadership skills."