In the days when journalism consisted of asking people what they thought rather than telling them what to think, Studs Terkel was a hero.

More oral historian than reporter, the Chicago-based writer and broadcaster used his weekly radio programme as the springboard for a series of remarkable books in which he asked ordinary Americans to describe their lives. Working, published in 1974, was one of those and its words form the basis for this musical by Stephen (Wicked) Schwartz, with songs and lyrics by a variety of hands. In 2011, the show was updated for a production directed by Gordon Greenberg, with new interviews to reflect the world of work and additional songs by Hamilton's Lin-Manuel Miranda.

It's an odd but sturdy construction, the scenes flowing easily into each other, held together by tenuous links and rather stronger themes. Despite the disparate jobs described – from a fireman to a waitress, to a press agent and a delivery boy - shared preoccupations emerge: the dignity of labour, the way we define ourselves through work, the freedom that even a menial job can bring, the oppression of factory workers, the way work's contribution to the fabric of society is overlooked.

There are some standout songs too. Miranda has contributed two corkers, a bright song for the burger flipper who finds liberation on the back of his bike, and a mini-masterpiece in the shape of "A Very Good Day", a duet about the people who care for those abandoned by their families, powerfully sung by Siubhan Harrison and Liam Tamne. Craig Carnelia's "Joe", the ballad of a retired man trying to fill his time, is wistfully performed by Peter Polycarpou. James Taylor proffers the heart-breaking "Millwork", a description of backbreaking, health-destroying boredom, beautifully sung by Harrison, and the lively "Brother Trucker", a paean to life on the road, vigorously sung by Dean Chisnall.

The cast perform each cameo with skill and commitment; Harrison is very funny as a frightened air hostess, Gillian Bevan conveys bitterness as a disillusioned teacher and enjoyment as the waitress who has turned the mundane into an art. Krysten Cummings makes the lament "Just a Housewife" (by Carnelia) both truthful and moving.

Schwartz holds the entire thing together with a couple of rousing anthems to the value of work and the way it makes America. Jean Chan's simple set effectively evokes both the world of labour (all distressed and oily) and Terkel's radio studio and director Luke Sheppard keeps things fizzing along through a brisk 90-minutes.

I was less convinced by Fabian Aloise's choreography and a staging that sets a chorus of young people in adoring listening poses as the oldies tell their stories. It felt frenetic and undifferentiated, flattening the effects of the very different tales. But with the help of Isaac McCullough's lively music direction, the entire show is full of energy and never less than engaging.

Working runs at Southwark Playhouse until 8 July.