I would add that the mother is urbane. She doesn't live in London but must live in a nearby suburb. She is only suspicious because she has seen too much of people in the big city. Like any other big city, London is full of people who live by preying...
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on others. She is not unkind, however. She actually offers to pay the old man's taxi fare without taking his umbrella. She finally takes it because he insists on giving it to her for one pound. You are perfectly right that she is curious. She follows the umbrella man for some distance and stands outside the pub spying on him, even though it is raining heavily. She is a pretty good mother. She takes her daughter to the dentist in the city and buys her a banana split afterward. She may seem prim, proper, and snobbish, but she is also humane. She seems to be trying very hard to instruct her daughter and to set an example for her to follow. The story is set in the right place, because England gets lots of unexpected rainstorms. The fact that the mother didn't bring an umbrella on this trip may be suggestive regarding her character.