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While it is true that Goedel's theorem applies to weak systems such as Robinson Arithmetic (and any decidable extensions there of), The proof of Goedel's result itself requires at least some amount of induction.

As a consequence the minimum system that Goedel's second incompleteness applies to is stronger than the minimum system that the first incompleteness theorem applies to.

I think if Gödels proof needs Q then that is OK.

Q cannot prove its own consistency. Which means there is no way of telling that Gödels theorems are proved in a theory that is inconsistent (where everything is true).

"it does not rely on any particular axiomatisation"

-- OK, what does it rely on then?

The axioms allowing one to express enough of arithmetic for Goedel’s methods to apply. As the comment says.