What is a good word for someone who "gets the joke"?

If your grandmother had keen mental perception and understanding of those things you mentioned, I would say your granny was "perspicacious"

• perspicacious - (adjective) "having or showing an ability to notice and understand things that are difficult or not obvious"

e.g.

• "She was much too perspicacious to be taken in by such a spurious argument."

• "The perspicacious plumber quickly detected the leakage."

• "Although my mom was a perspicacious woman, she was not able to understand why my brother had lied.

• perspicacity (noun) "the ability to understand things quickly and make accurate judgments"

  "Grandma was a woman of exceptional perspicacity."