First author researcher on a manuscript left job without publishing
If the slightly outdated paper comes back from review with a decent amount of edits to be made, potentially new portions of analysis, while the first author states that they do not have time to work on the old manuscript, can one of the co-authors potentially take over as first author?

Potentially, yes. Far from a guarantee. Probably the old first author would become second or co-first, depending on norms in your field.

Does the original first author have to give consent?

Yes. In fact, everyone has to agree to the new order, even if their relative position is unchanged.