How far would you go for family? That’s the deceptively complex question posed by “The Winslow Boy,” a satisfying, very British drama from the Roundabout Theatre Co. now open at the American Airlines Theatre. In the case of the Winslows, an upper-class clan living in London’s Kensington district, the answer is: you go until it hurts, and then you go some more.

It’s a time just prior to World War I, and young Ronnie Winslow (Spencer Davis Milford), a cadet at the Royal Naval College in Osborne, has been expelled, accused of stealing a 5-shilling postal order. So afraid is the teen of telling his father, Arthur (Roger Rees, of TV’s “Cheers” and “The West Wing”), that he cowers in the backyard rather than convey the news.

Turns out, Ronnie needn’t have been put off. With the empathy of a fairy-tale parent, Arthur Winslow asks his son if the accusation is true and accepts the boy’s adamant denial, vowing to clear his name as a matter of principle.

All this transpires within the first moments of Terence Rattigan’s thought-provoking play, first produced on Broadway in 1947, a year after its London premiere. The narrative is based on a real-life episode that took place in 1908 at the Osborne academy. This new Roundabout production, helmed by Lindsay Posner, was produced last season by London’s Old Vic.

“The Winslow Boy” is set entirely in the family living room, ornately decorated in pleasant greens by Peter McKintosh (“The 39 Steps”), though the action -- the alleged theft and subsequent courtroom drama -- occurs offstage. Characters often retreat, somewhat inelegantly, to an unseen back room, allowing privacy to their counterparts downstage.

Arthur is driven by a belief that his son was discharged without fair representation: “An injustice has been done,” he tells wife Grace (Mary Elizabeth Mastrantonio, not seen nearly enough). “I am going to set it right, and there is no sacrifice in the world I am not prepared to make in order to do so.”

It’s a credit to actor Rees that the dated storyline and Masterpiece-esque class distinctions don’t come off as exceptionally antiquated; nor does Arthur seem imperious, even while busying the House of Commons with a battle over “a bally postal order” as war threatens to rage across Europe.

The story takes an added urgency with the entrance of an arrogant barrister, Sir Robert Morton (the enigmatic Alessandro Nivola), whom Arthur and daughter Catherine (Charlotte Parry, smart) manage to attract after losing an appeal.

As the first act concludes, Sir Robert interrogates Ronnie in front of the assembled Winslows. The family is aghast at the aggressive examination, an adult shredding an adolescent in a living room that has become a makeshift court. It’s an edge-of-your-seat moment in a play paced more quickly than some recent Roundabout productions, but still lacking substantial tension. The family’s willingness to allow Sir Robert free reign pays off, and he agrees to take on the boy’s cause in Parliament.

Joining the mix is fellow solicitor Desmond Curry (Michael Cumpsty, a Tony nominee for “End of the Rainbow”), a longtime family friend more meek than Sir Robert who has long harbored feelings for Catherine. Cumpsty makes a big impact in a relatively sparse role, particularly with an overture to Catherine toward play’s end that is tragicomic and bittersweet.

As Sir Robert and Arthur press forward with the case, there’s a toll on the family. Catherine’s engagement to her fiance John (Chandler Williams) is threatened. Older son Dickie (Zachary Booth) must drop out of Oxford, owing to the expense of Ronnie’s defense. Grace contends with the prospect of losing Violet (Henny Russell), the parlor maid who spills family secrets with alarming frequency.

Each Winslow accepts fate with a stiff upper lip. As for Milford’s young Ronnie, we take his word from the outset. The young actor doesn’t give us much reason to doubt his innocence, though we’re certainly left guessing what Parliament will say about the issue.

Fret not: there is an unambiguous resolution to matters. When it comes, Ronnie isn’t in court. He’s at the movies. The boy has his own sense of just where the line is between family devotion and pragmatism. The principled pursuit of this particular white whale is left to his elders.

“The Winslow Boy,” through Dec. 1 at the American Airlines Theatre, 227 W. 42nd St. Tickets: $52-$137. Call 212-719-1300 or visit roundabouttheatre.org.

Follow Robert Kahn on Twitter@RobertKahn