The star, 62, who grew up in Dudley, said the “series will be a tribute to the giants who came before us and walked cold streets to create new lives for themselves”.
Fictional series Three Little Birds will air on ITV, following the adventures of gregarious sisters and a bible-loving acquaintance as they board a ship heading for a new life.
Sir Lenny said: “My mother spoke often about how difficult life was back in the day when she first arrived in Britain to begin her new life. The stories she told, and indeed the narratives my brothers and sisters beguiled me with, over the dinner table made me think about writing a fictional account of three Caribbean women.
“They all get to the UK with one thing on their minds – a new life.
“Although these are fictional accounts my mother’s narrative will run throughout these stories and hopefully the stories of other post-Windrush arrivals will trigger memories, smiles and tears too.”
ITV’s head of drama Polly Hill said: “I fell in love with (characters) Leah, Chantrelle and Hosanna the moment I read Lenny’s script and I think the audience will want to go on their journey with them.
“Lenny has written a funny and moving, but most of all life-affirming, drama that is a tribute to his mother and all those who came to Britain and made it their home.”
Scriptwriter Russell T Davies will be an executive producer on the drama alongside Sir Lenny.