I've sent my own leaflets to the stage managers of the Palace theatre (for Spamalot) and the Noel Coward (for Avenue Q).
Both came back signed by some of the cast
so I guess if you have something to send its worth a shot!

While I dont know if it would work for musical theatre, I did send a request letter to Annette Badland when she was in The Vortex, a play in the West End. Along with the photo I sent her (which she signed) I asked for a cast signed flyer if possible, and she sent one which was indeed signed by the entire cast. This is not the same as the American system, where they will send a playbill (often without you even requesting it). They wont send you a programme for free, but if you ask they might send a flyer (usually A4 folded 3 times, or smaller). E.g. you could write to "Mamma Mia! Cast, c/o Stage Door" to the theatre, and chance your luck.

Absolutely NOT. West End does not do ANYTHING similar to the US/Broadway style of autographs.

I've only seen two cast-signed successes, and both times the item (a foldable poster) was sent by the person seeking the autogragh (Phoebe on here may be able to help you).

For Mamma Mia!, current star, the beautiful and very talented Linzi Hateley is extremely good with fanmail. She sent me a flier of the show and a headshot when I wrote to her (and signed a picture of her as Carrie White that I sent !).

You can certainly try. You may see a different outcome than I have. I've written four times, each time sending a SA(S)E (or IRCs), with $USD for a signed programme/something, and each time I've gotten nothing. I've written to: Phantom of the Opera, Les Miserables (x2), Gone with the Wind: The Musical, and the EVITA revival. Nothing..!

If you're from the US, you could always try to send to the star of the show you saw (and potentially send $USD) for a signed programme by them (and they usually send out fliers). Ruthie Henshall .. I sent twice (by accident actually). Once, when I didn't send the $USD, she sent me a headshot and a flier for her then current show, Marguerite. The other time I sent $USD (about $6-8) and she was able to get me a programme and sign it.

If you need any other help, PM me or possibly Phoebe -- hopefully I'm not putting too much on her but she does have the experience with the cast-signed Joseph.. poster!

Okay, there are two West End performances in London that I have seen.
They are the Lion King and Mamma Mia.
I would really like a signed leaflet or something by the cast.
Would it be advisable to send letters to the stage managers of the West End?