Can a strong nucleophile carry out an SN1 reaction?
You've pretty much said it all already. SN2 is ruled out at the tertiary centre. Since the attack of the nucleophile in the SN1 mechanism is not rate-determining, it doesn't matter whether the nucleophile is weak or strong; iodide works fine.

In this case, it's more likely to get intercepted by the solvent itself, which is present in much larger quantities and is also nucleophilic, but that's another story.