"A one minute talk" is a different scenario from "a one minute's talk".
"A one minute's talk" is wrong.
In English, most singular nouns need a determiner of some kind. Usually these are the articles: "a book", "the book". Sometimes they're demonstrative adjectives: "this book", "that book".
The cardinal number "one" can also be used as a determiner: "one book".
We can also use possessives as determiners: "Bob's book".
Thus, all of the following are correct:
- "I bought a book."
- "I bought the book."
- "I bought this book."
- "I bought that book."
- "I bought one book."
- "I bought Bob's book."
However, each noun can only have one determiner at a time. I wouldn't be able to say "I bought a one book." "A" and "one" are both determiners, and are both referring to "book". You can only have one determiner per noun.
In the phrase "A one minute's talk", "one" acts as the determiner for "minute", and "one minute's" (as a possessive) acts as the determiner for "talk". The "a" is redundant. Thus, you want "one minute's talk".
By comparison: "A one-minute talk" is right.
I can take the phrase "one minute", and treat it as one big adjective. (I would usually indicate this by connecting them with a hyphen - "one-minute".) Thus, I could say:
- "a talk"
- "a boring talk"
- "a one-minute talk"
- "a longer-than-necessary talk"
All of these are correct, because "a" is the determiner for "talk", and "boring", "one-minute", and "longer-than-necessary" are all behaving as adjectives modifying "talk".