Allison, one of the most highly-respected engineers in Formula One who was in his third spell with the Enstone-based team, has been instrumental in Lotus' strong form to date in 2013 but has walked away from his role ahead of the start of the European season in Spain this weekend.
Chester steps up from the role of engineering director to fill the void, with Allison's next move still uncertain.
Ferrari are believed to be leading the chase to snap him up, although rivals McLaren are understood to have made an approach. Mercedes and Williams are also known to be interested.
Team principal Eric Boullier said: "As a team and individually, we would all like to thank James for his efforts during his three stints at Enstone, and wish him all the best in his future endeavours."
On Allison's successor, Boullier added: "Nick is well known to everyone at Enstone having been with the team for over 12 years.
"He is already directly involved with this and next year's cars, ensuring a smooth transition which has been under way for some time.
"It's an illustration of the strength and breadth of talent at Enstone that we can draw on personnel of the calibre of Nick.
"He assumes his new position at a tremendously exciting time for the sport. The 2014 technical regulation changes present many challenges, while our current position of second place in both the constructors' and drivers' world championships mean we cannot lose sight of this year's development battle.
"Nick really has his work cut out, but we know he is more than capable of handling the tasks ahead."
Chester moved to Enstone in 2000, initially as a race engineer before taking on the roles of head of vehicle performance group and head of performance systems.
"I'm grateful to the management at Enstone for the faith they have in promoting me to this position.
"I am very aware of our need to keep pushing development of this year's car whilst developing next year's to a set of very different regulations.
"There are some exciting times ahead for Enstone and I'm honoured to be part of it."
Allison's association with Enstone dates back to 1991, when he joined Benetton's aero department. He the team's head of aerodynamics in 1995 after returning from a stint at Larrousse, before leaving to oversee Ferrari's trackside aerodynamic operations in 2000.
He return to Enstone for a third time in 2005 with the team now renamed as Renault, and as deputy technical director played a key role as the team won back-to-back drivers' and constructors' titles in 2005 and 2006.