Mean Girls and Frozen on Broadway: we're in for treats in the West End

As somebody who lives and breathes West End theatre, wandering around New York's theatre district could be a confusing experience. Phantom, The Lion King, Wicked, Hamilton, Aladdin, The Book of Mormon, Harry Potter, Chicago, The Play That Goes Wrong, Kinky Boots, The Ferryman, Network: which city am I even in?! Hang on, those last five have closed in London, so…I must be on Broadway! Even so, with Come From Away and Waitress starting West End previews, and a London Dear Evan Hansen about to go on sale, the theatrical landscapes on opposite sides of the pond have seldom looked so similar.

Although no official announcements have been made, many are hoping that at least two more current Broadway blockbusters will be making the transition from Great White Way to West End. Golden-touched Sonia Friedman, whose current portfolio includes the upcoming star-studded All About Eve, plus the global iterations of the Potter plays and The Book of Mormon, is one of the principal producers of Mean Girls, which plays to standing room only nightly, and the word is that they are just waiting for the right London house to become available. Meanwhile, the stage adaptation of Disney's Frozen, which actually has a British direction and design team – Michael Grandage and Christopher Oram respectively – is hotly tipped as the re-opening production next year at the drastically remodelled Theatre Royal Drury Lane.

When Frozen needs to deliver, it surely does

I think Frozen may play even better in the grandiose surroundings of the Lane than it does in its present Broadway home, the attractive but less imposing St James, whose illustrious history includes the original runs of such American classics as Oklahoma!, The King And I and Hello Dolly! It's a sombre spectacle is this icicled fairytale: opulent, atmospheric and almost Ibsen-esque in its exquisitely rendered gloom, it only fitfully bursts into the colourful life you might expect more of in a Disney show. At the packed Tuesday night performance I saw, the families in the audience responded to these light-filled respites with roars of approval, as they did to the eight numbers held over from the movie soundtrack. Personally, I was more intrigued and excited by the darker elements in both the story and the often beautiful staging.

Terrific as they are, there is nobody in the Broadway company that couldn't be replaced more than adequately with UK talent, although it must be said that Caissie Levy (as Elsa) delivers a roof-shaking "Let It Go" at the end of the first half that wipes the floor with the soundtrack version. Grandage is economical with the magic but when the show needs to deliver, it surely does.

Given the huge success of Heathers last year and the excitement surrounding the London Waitress this year, I imagine that a West End version of Mean Girls will clean up, not least because it's source material – Tina Fey's gleefully sour takedown of über-popular teen queens and the high school kids who fear them – was a far bigger, and more recent, movie hit. Also, the stage musical version is just so much damn fun.

Fey brings the same piercing, but occasionally savage, wit and kookie eccentricity that characterises her TV comedies 30 Rock and Unbreakable Kimmy Schmidt to the stage, while the score (by Fey's real-life husband Jeff Richmond, with lyricist Nell Benjamin) is an earwormy confection encompassing bubblegum pop, classic Broadway ballads, and an epic James Bond-like sweep. Casey Nicholaw (the director-choreographer behind The Book of Mormon and the London Dreamgirls amongst many others) helms an oil-smooth, candy-coloured production shot through with moments of improbable, hilarious invention (until you've seen a pair of dancers transform into model cars you haven't lived!).

Again, this is a show that London casting directors will have a field day with, although I rather hope Ms Friedman uses her clout with UK and American Equity to get Grey Henson – stunning as the heroine's "almost too gay to function" best friend – to open the West End version. Like a young Nathan Lane, he is an authentic triple-threat with buckets of charm, tossing out killer one-liners or insouciantly delivering complex tap breaks all the while with the bewildered expression of a man who never really meant to be there. He is priceless.

The Cher Show turned out to be rather more classy and thought-provoking than I'd expected

No word yet on a London transfer for one of the biggest hits of the current Broadway season (Frozen and Mean Girls both opened in the last one), The Cher Show, a lavish, ultra-camp, entirely joyous stage bio of, well, Cher obviously. I assumed it would be a lot of fun (and it was) but it turned out to be rather more classy and thought-provoking than I'd expected. It screams Palladium.

I'm also hoping for a West End mounting of new musical The Prom – again staged by Nicholaw, and from the creative team behind The Wedding Singer and The Drowsy Chaperone – which combines heartfelt coming-out story, sheer Broadway pizazz, and a heartwarming "Love Is Love" message that feels very timely. It's irresistible, surprisingly moving, and the funniest musical I've seen since The Producers.

In the meantime, who do we think the West End Anna and Elsa will be? Or the London Regina George…? I know who I'd cast…