It might be argued that Cambridge Dictionary licenses this usage (though POS assignment may well be contested):
after [adverb] A2
later than someone or something else:
- soon after Hilary got here at midday and Nick arrived soon after.
- I can't go next week – how about the week after (= the following week)?
- [not standard] She got back at 4.30 and went to see Emilie after (= after she got back).
[highlighting mine]
However, there is a retrievable 'that' / 'that one' / 'next week' in the relevant example given in CD, with antecedent 'next week' earlier in the sentence. Hence Jespersen and others see 'after' here as better classified as an intransitive preposition.
With He said there would be a show the Saturday after, there is a lack of antecedent. The Saturday after what? In my opinion, this somehow seems more of a problem than with 'following' in 'He said there would be a show the following Saturday', where the default interpretation is 'following the time when 'he' was speaking'.
I'd say John mentioned that the show scheduled for the last Saturday in April had been cancelled. He said there would be a show the Saturday after is again acceptable ... though probably not the most idiomatic choice.