A few experiences here leap to mind -
1/ An actor (1 of 3) was 'let go' late on the Saturday before the Wednesday Preview. Sunday was the Day Off with Monday and Tuesday being the Tech Days. Management found an actor who had done the role, flew them in and on the Monday morning I and the other two cast walked the actor thru the blocking. Rest of the Tech time went as originally planned and we Previewed without issue or comment to the audience. There had been talk of putting an insert in the program noting the swap, but as the program had not been put to bed until Monday, I pushed for simply swapping out the photo and bio, which was done.
2/ While on an extended tour, one of two two male leads collapsed backstage during a performance. When it became apparent he was not going to be able to go on, I turned to one of our other actors (small role and finished at this stage) and asked if he would be willing on going on (with book), if required. He agreed. Ill actor was put in civvies and resident theatre management saw him to hospital, the other actor got into his costume, I got word to actors onstage there was a problem and, at the actor's intended entrance, I came onstage (in costume as all scene changes in full light), stopped the show and explained we had a had a ill actor backstage, we had a 'volunteer' willing on going on in the true 'The Show Must Go On' form and asked for an applause poll as to whether we would stop the show or continue on. The audience overwhelming voted to continue on and after a moment's prep, we continued the play. The replacement remarkably knew a lot of the lines (rather by osmosis after 24 weeks with the show) and was gently guided about onstage by fellow actors, if needed. Audience went wild afterwards.
Lead actor ended up being out for two days and we did an emergency rehearsal the next day with our 'volunteer'. He ended up doing both his original role and that of the Lead with the only troublesome moment a point when both characters were blocked to be onstage at once. I resolved this by having one character in doorway, facing US, speaking the lines and replying in the other character's voice!
3/ Lead actor (musical) whose voice suddenly went during Act 1 of 2 act production and who said at the Intermission she would be unable to continue on. I had already put a call into management at soon as I noted problem in Act 1 and so the AD was in the theatre by Intermission. Upon being informed she could not continue, I let AD know, he conferred with actor and decided that he would cancel the show. At the end of Intermission, he came onstage and noted the problem and the cancellation, along with an invitation to return for another performance.
Bigger problem was that actor did not go to theatre doctor as was arranged, but disappeared out of the theatre during AD's speech and did not answer calls to cell or hotel, so no way of knowing what the situation was for upcoming shows. Management dealt with this and she returned the following evening.
4/ In the UK, it is often the case there is a Stage Manager and a Deputy Stage Manager (and then various ASMs) and it is often the DSM who will be responsible for taking down blocking in rehearsal and for calling the show. On one production I was involved in as SM, my DSM reported feeling very unwell after the show and, whisking her to hospital, we discovered she had appendicitis and would need immediate surgery.
I called an emergency rehearsal for the next day so that I could run thru the show with the Book. Just as well as her Calling was noted in a form totally unfamiliar to me, in pencil, in microscopic print - and the show was very intensive with visual Qs called almost blind without benefit of monitor or clear line of sight. I rewrote the Qs in BIG BOLD PRINT and inserted these into the Book between rehearsal and evening performance, my ASM took up some of the slack left by me 'in the chair' and the Company muddled thru. A replacement DSM came in the following week (the original out of commission for 3 weeks) and this person was very pleased to see my additions to the Book.
Which only begs reminder that those who Call a show remember they too are not immune to 'Life' and the Book should should be able to (within reason) be understood by anyone in an emergency...