Based on Robert James Waller’s bestselling novel The Bridges of Madison County, Jason Robert Brown’s musical, which had a short run on Broadway in 2014, needed either to rise above sentimentality or to embrace it unashamedly. Trevor Nunn’s brilliant and involving production does both. I had to pinch myself to check that I was, against expectation, enjoying myself inordinately, and to accept that even the songs with the lamest lyrics were turning out to be no obstacle to a great night out (handkerchiefs are required).

This is the story of a middle-aged Italian immigrant, Francesca, who lives in Iowa and is married to a former American soldier, now a farmer. Francesca is outstandingly played by Jenna Russell, who makes you sympathise with each lilting Italian syllable that she says and sings (the show is uneven musically, but there’s a touch of Joni Mitchell to its best numbers). When Francesca’s husband and children go off to a county fair without her, she looks forward to lie-ins and iced tea and studying seed catalogues. She is not, you would think, the sort of woman to stray.

But when Robert Kincaid, a photographer on an assignment to photograph the bridges of Madison county for the National Geographic (played in the 1995 film version by Clint Eastwood opposite Meryl Streep), drops by asking for directions, it turns out to be the heart’s directions that count. Robert is a free spirit (and, like most free spirits, essentially trapped). Edward Baker-Duly steps into Eastwood’s shoes effortlessly – he even looks a bit like him. He lopes about with casual ease, more cowboy than cameraman, sings with a fine tenor voice, and a romance is cooked up along with the vegetable soup they prepare together. It’s cosily improbable but who cares? The fennel now chopped, we are idling towards a moral climax: should Francesca throw away family life for new love?

Her trusting husband is nicely played by Dale Rapley, and there are decent performances as their feistily self-involved children from Maddison Bulleyment and David Perkins, and enjoyable stretches in which bickering family talk turns into song. Add in starry singers Shanay Holmes and Gillian Kirkpatrick and the recipe is complete. Holmes is bewitching as Kincaid’s young ex-wife (haunting an early scene), and the way Kirkpatrick switches from spying neighbour to siren singer is theatrical gold – Nunn at his innovative best. Jon Bausor’s wonderful set creates huge spaces within the Menier’s confines, starting with screened projections of the sea that Francesca crossed to get to Iowa which turn into blurred prairie: “300 acres waiting to be tamed”.

In a week of heatwave and American musicals, crossing from Iowa to Oklahoma! was a seamless transition. It is one of the strengths of Rodgers and Hammerstein’s 1943 musical that, in spite of its greatest hit, Oh, What a Beautiful Mornin’, beautiful days are not equally beautiful for everyone. In Jeremy Sams’s exuberant and sensitive production, Jud Fry’s death tarnishes Curly and Laurey’s wedding day and Aunt Eller’s motto – “You’ve got to be hardy” – rings true. As the aunt, Josie Lawrence has perfect timing. When Laurey dismissively pushes Curly on to a hay bale, Aunt Eller says, after a pause: “She likes you.” Roles that make fun of age are tricky but this Aunt Eller is always ready to take another swig at youth, and Lawrence satisfactorily shows how older, wiser and more foolish can come together.

The rural set on the wide Chichester stage is pleasingly understated (designer Robert Jones), with bales and baskets and room for manoeuvre. Hyoie O’Grady is a charming Curly with a soaring voice, and Amara Okereke’s lovely Laurey sings with sincerity. They perform People Will Say We’re in Love exquisitely, with the ardour and irony the song requires.

There is phenomenal support from the rest of the cast, especially from Isaac Gryn, who plays Will Parker. Still cooling his hot heels at drama school, he’s a born performer (even a dab hand with a lasso). Emmanuel Kojo’s Jud is superb too – earthed yet emotionally precarious, with a handsome singing voice. Scott Karim’s Ali Hakim, an exotically dressed mountebank, has irresistible eccentricity as he initiates us into a “Persian farewell” (no spoilers here).

Matt Cole’s fresh choreography uplifts – in particular when, with a swoop of a cowboy’s hat (Gryn again), the company tilts in the hat’s direction. This is a musical filled with projections and imagined transport (along with the “Surrey” that eventually materialises), and Chichester’s splendid production reminds us what a road-holding vehicle Oklahoma! is.

The problem with Blues in the Night is that there is no vehicle to carry the songs of Duke Ellington, Bessie Smith and all. To some extent this does not matter: you can’t go wrong with mighty singers such as Clive Rowe and Sharon D Clarke. But the almost nonexistent narrative becomes an issue (you see why even the shaggy stories that keep operas afloat are necessary). Conceived by Sheldon Epps, the show started on Broadway in 1980 and is revived by director Susie McKenna, who tries to coax songs that were not written for the purpose to communicate with one another.

The set (Robert Jones has been working overtime) is a convincingly conceived but congested hotel with a single neon light and barely enough room for the nifty dancers (Aston New as the Hustler and Joseph Poulton as the Barman). Each female singer is stuck in her room, only intermittently spilling out in the direction of a central piano. Being static is appropriate to songs about loneliness but does the show no theatrical favours.

And how blue do you want your blues? Do you ever pine for another mood? Clarke moves gloriously, often slow as molasses, owning the stage, and goes for the darkest shades in Wasted Life Blues (a stunning performance). Rowe is victoriously involved in the struggle not to let sad songs turn comic – his is a royal blue as he takes things in his stately stride. Debbie Kurup gets a Jean Rhys-like melancholy into her pieces and Gemma Sutton holds her own with courage and intelligence. Their songs are paler blues, perhaps. Finally, three cheers for pianist and musical director Mark Dickman and his flying fingers – his blues are not blue at all.

Star ratings (out of five)
The Bridges of Madison County ★★★★★
Oklahoma! ★★★★
Blues in the Night ★★★

• The Bridges of Madison County is at Menier Chocolate Factory, London, until 14 September
• Oklahoma! is at Chichester Festival theatre until 7 September
• Blues in the Night is at Kiln, London, until 7 September