The 90-minute drama includes Paige as Mistress Quince, Lucas as Bottom, Hannah as Theseus, and Hamlet actress Maxine Peake as Titania.
Adapted by former Doctor Who writer Russell T Davies, the 90-minute drama will also star Richard Wilson and Bernard Cribbins, alongside Eleanor Matsuura, Hiran Abeysekera, Fisayo Akinade and Javone Prince.
Commenting on the casting, Davies said: "This is such an exciting range of actors from stage and screen, from wild comedy to high drama, and some making their first ever appearance on camera. The dream is coming to life like never before."
The adaptation is due to air on BBC One in 2016 as part of the BBC’s Shakespeare season, which will commemorate the 400th anniversary of the playwright’s death.