I believe they did an extended run on this production over a period of time. There is an outstanding chance this was signed there. I personally don't think someone forged this and then put a wrong date on the page.
My personal belief is if you have doubts about a signature it's better to pass on it and wait for an example you are comfortable with.
From the British Universities Film & Video Council website:
C%26C%201951%20II.png (this is an archive document that shows the BBC did a radio recording of the play, at the theatre, 3 days before the date on the signature)
From Vivien, The Life of Vivien Leigh; biography by Alezander Walker:
I was not suggesting forgery with the dates - I was stating I could not find it. This Leigh - she is normally rapid and light to start - I would look for a better, more typical example were it me. I just sold my 1949 Streetcar set and replaced it with a signed McBean photo of Leigh as Blanch .