Peake will star as Titania, Lucas will be Bottom and stage star Elaine Paige returns to TV as Mistress Quince.
The former Doctor Who writer has written what is described as a "bold and accessible" adaptation of Shakespeare's play.
It will be shown as part of a new Shakespeare season on BBC One in 2016.
Bernard Cribbins and Richard Wilson will also star in the adaptation as Snout and Starveling respectively.
Also appearing in the cast is John Hannah, as Theseus.
PhoneShop's Javone Prince will play Snug and Fisayo Akinade who starred in  Russell T Davies' Banana and Cucumber will play Flute.
Maxine Peake played Hamlet on stage last year, saying she took the role of the vengeful prince because of a scarcity of great parts for women.
She added she found Shakespeare's female roles "quite problematic".
Russell T 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 described as having "as much attitude and invention as any theatrical interpretation".
It is being aimed at all age ranges, said the BBC, from "children, who can laugh at Bottom and his Mechanicals and marvel at the fairies' awesome powers" to "adults, who know those broken hearts and star-crossed lovers all too well" and "whole families, united in front of the television to enjoy the play's dazzling world of danger, jokes, scares, poetry, thrills and fun."

Two women aged, 18 and 19 from Grangetown, and a third aged 19, from London, have been charged with fraud by false representation.
The arrests were made after people were approached by cold callers who claimed to be collecting money for Unicef.
But the charity confirmed no one had been authorised to fundraise in the area.
The women appear at Cardiff Magistrates Court on Friday 1 May. Two other men who were also arrested have been released on bail pending further investigations.

Shared hallways and easily accessible rental properties mean tenants' details are easier to steal, according to credit reference agency Experian.
This group also tend to use mobile and online services more often, offering more opportunities for thieves to find personal details.
Another recent survey found that ID fraud accounted for 41% of all fraud.
In March, fraud prevention service Cifas said that digitally-savvy young adults were increasingly becoming targets, at fraud hotspots in London, Leicester, Birmingham, Manchester, Leeds and Glasgow.
Now, Experian has said that identity theft has become the most prevalent form of fraud for the first time since the recession.
The victims in one in six of these cases were young tenants, it said.
Wealthy couples were next on the list of victims, with this group particularly targeted in card and loan fraud.
"[The findings] highlight the need for people to think about how they protect their personal information both on and off line," said Nick Mothershaw, of Experian.
High Street banks and the police are running a campaign to make customers more aware of the threat of fraud.

Megan Lee, from Oswaldtwistle, Lancashire, died on 1 January, two days after she was admitted to hospital.
Royal Spice has been temporarily closed by Hyndburn Council due to below standard allergen management and a mice infestation.
Two men from the takeaway held on suspicion of involuntary manslaughter by gross negligence have been bailed.
Blackburn magistrates granted the Hygiene Emergency Prohibition Order earlier.
The authority said the premises in Union Street, Oswaldtwistle, will remain closed until it was satisfied health risks had been removed.
Police have said a post-mortem examination had been carried out but the full results would not be known "for some time".
The arrested men, aged 37 and 38 and from Rossendale and Blackburn respectively, have been bailed until 7 July.
Maxine Peake, Matt Lucas and Elaine Paige are heading the cast of Russell T Davies' adaptation of A Midsummer Night's Dream.
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Three people have been charged over allegations of fraudulent fundraising in Cardiff.
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Renters face the greatest risk from identity thieves owing to their domestic set-up and lifestyle.
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A takeaway has been shut down following a teenager's death from an apparent allergic reaction to one of its meals.