Would I be fine, even if I didn’t ask for their OK?
Cheers 
In terms of copyright, it’s important to distinguish between the recording and the interview itself. You can have copyright on a recording. You can’t have copyright on something you said in public (in itself).
Whoever owns the copyright on the recording of the original interview has the legal rights regarding its use (outside of “fair use” exemptions). Just because the interview is “public” doesn’t mean anyone can use the recording of it (unless the legal copyright holder has expressly given up their rights).
So for instance, if Will Smith is giving an interview to the BBC on a public street, you can not (legally) use the recording in your project without the BBC’s permission - because they own the copyright on it. However, if you snuck in beside the BBC reporter and made your own recording of the exact same interview, you could use it fine, because you own the copyright on it. Will Smith doesn’t get any say in either scenario, because he’s willfully speaking in public, and doesn’t have a reasonable expectation of privacy.
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…taken out of context, spoken words in public, not wriiten down anywhere are pretty safe and common ground…