The one-off commission, titled Bette and Joan and Baby Jane, will tell the story of the resentment which allegedly grew between Crawford (Oberman) and Davis (Tate) when they filmed their 1962 horror movie Whatever Happened To Baby Jane?.
Bette and Joan and Baby Jane was written by Oberman, best known for playing Walford's Chrissie Watts, before being brought to the BBC by the independent production company CPL.
CPL has also secured three other drama slots on Radio 4 under a new system which saw selected suppliers invited to pitch for BBC airtime.
The company's managing director Danielle Lux told The Guardian: "We were delighted to secure four Afternoon Play commissions via the new Radio 4 batch tender system.
"And to have talent of the calibre of Tracy-Ann Oberman and Catherine Tate involved in the first of these plays has been a wonderful start for CPL."
Bette and Joan and Baby Jane will air on April 29.