When and how are Tom and Becky discovered missing in The Adventures of Tom Sawyer?

Becky and Tom are discovered missing when they don't show up at church the morning after Saturday's picnic outing. The adults, each one thinking the children slept at someone else's house, compare notes. They then realize that the twosome slept at nobody's house.

At this point, the townspeople realize that Tom and Becky have disappeared. Nobody saw them on the ferry coming home. The assumption is that the children are probably lost in the caves they were so happily exploring during the picnic. The adults set out to search the caves and find the missing children.

The townspeople are correct that Tom and Becky are lost in the caves. However, despite searching and shouting, they are not able to locate them. As time passes, the fear grows that the two are dead.

As we can see, child safety standards were not what they are today. Today it would be extremely unlikely for children to be allowed to explore caves on their own. It would also be highly unlikely that a ferry would take off without a head count to make sure all the children are on board. Modern parents would also almost certainly double check to know where their children are sleeping.

Tom and Becky are not discovered missing until church the next morning.  Although they did not come home after the outing the day before, their respective guardians both assume the children had spent the night at Mrs. Harper's.  Noticing that Becky is not at church, Mrs. Thatcher asks Mrs. Harper, "'Is my Becky going to sleep all day?  I just expected she would be tired to death'".  Mrs. Harper responds with surprise, not having seen Becky, and then Aunt Polly approaches her with a similar question about Tom.  The Harpers, Mrs. Thatcher, and Aunt Polly are by this time understandably worried, and ask the children and teachers at the Sabbath-school if anyone knows where Tom and Becky might be.  No one had noticed if Tom and Becky were on the ferryboat on the homeward trip the night before, and one man suggests the frightening possibility that they might still be in the cave (Chapter XXX).

When do Aunt Polly and Mrs. Thatcher discover their children are missing in The Adventures of Tom Sawyer?

Aunt Polly and Mrs Thatcher realise this when they meet each other and Mrs Harper in church the day after the picnic. This is described in Chapter 31. Mrs Thatcher assumed that Becky didn't come home after the picnic because she stayed overnight at the Harpers, but Mrs Harper tells her she didn't. Things are made worse when Aunt Polly comes to ask whether Tom stayed with either the Harpers or the Thatchers overnight, as he hasn't come home either. This is the first time that people realise that Becky and Tom are missing, and fears are raised that they got left behind in the caves during the sightseeing expedition that followed the picnic.