The 21-year-old will star as a young woman whose fiance is serving in Afghanistan and the experience brings her closer to her great grandmother (Syms) in new drama Timeless.
It is part of a new series of one-off comedies and dramas for the Playhouse series on Sky Arts which also features actors such as Friends star Matthew Perry, comedienne Jo Brand and Ben Whishaw, who starred as Q in Skyfall.
Delevingne has been trying to launch an acting career with a brief role in the 2012 screen adaptation of Anna Karenina and a forthcoming role in Michael Winterbottom thriller The Face Of An Angel.
Timeless has been scripted by Tim Firth, who was the writer behind movies such as Calendar Girls and Kinky Boots, and will be screened on June 19.
Perry will appear in a film written and directed by Jon Ronson called The Dog Thrower which features a soundtrack composed by enduring indie act Belle & Sebastian.
Whishaw features in a story called Foxtrot - created by playwright Polly Stenham - along with Billie Piper about an ill-fated heist by a girl gang.
And Brand stars in the comedy Damned - which she co-wrote with Morwenna Banks - set in a social services department.
Other films in the new Playhouse series feature Simon Callow, Jane Horrocks, Daniel Mays, and Mark Strong.
Director of Sky Arts James Hunt said: "We're delighted that such outstanding writers, directors and actors, all at the very top of their game want to come and 'play' with us. "