Highly subjective question; If you were to see only one West End musical in London, would you prefer 42nd Street or An American in Paris?

Six of one/half dozen of the other really . . . Which show interests you most?

My sense would be to avoid the America-style musicals and see something more British. After all, you came all that way to be in London.

We loved <em>The 39 Steps</em> and <em>The Curious Incident of the Dog in the Nighttime</em>, but I'm not sure either is still running.

ssander

I don't know about that necessarily. I've seen singing in the rain and guys and dolls in London and they were absolutely wonderful ( have also seem the 39 steps and the curious incident )

If you are looking for an authentic London theatrical experience, neither qualifies -- technically. Both are drawn from American movies, both have music by good (42nd Street) and great (American in Paris) US composers. 42nd Street has enjoyed a longer history on the live stage.

So what? London has for decades been a source of musicals that cross the Atlantic to North America, so vice versa is fair play. Only experts can differentiate between current national musical theatre traditions.
To be truly authentic in London might require finding a production of Gilbert & Sullivan (which I would gladly do) but could befuddle those not accustomed to the operetta style. Wikipedia is very helpful on the histories of the two shows you mention.

For sheer frivolous fun I would try out 42nd Street. I love Gene Kelly in the movie of American in Paris but the story is uneven and, from the reviews, that remains a challenge for the stage. But hey, a couple of hours of Gershwin songs is always time well spent.
Then do go to a British comedy and figure out the jokes (you probably will laugh hard even if the nuances slip by.)