We don't have an official policy on the use of naturals with Sagittals, but if we did, I think that Sagittal symbols (both pure and mixed) should be treated the same as conventional sharps and flats. Therefore, Sagittal accidentals are always sticky for the remainder of the measure, and they may also be used in key signatures.
However, there is one issue to consider: in conventional music, if there is an F# in the key signature and an Fx accidental is required, I have seen a natural sign placed to the left of the double-sharp accidental to cancel the sharp in the key signature. I didn't think that this is really necessary, because I thought that a double-sharp by itself would be sufficient to cancel the sharp. Similarly, I would think that, if there were an F# in the key signature, a single-shaft Sagittal accidental placed to the left of a note positioned on "F" (or "F#") would be sufficient to cancel the sharp in the key signature, because had one desired instead to retain the sharp and add the Sagittal accidental, then it would have been simple (and unmistakably clear) in the mixed version to include the sharp beside the Sagittal accidental, or in the pure version to use the multi-shaft equivalent. Thus a natural sign would be used only to indicate a note without accidentals of any kind.
Others are invited to comment!