My partner and I play 2NT-3NT shows 5♠s and 4♥s (unwisely it seems).  I obviously forgot until partner alerted.

The 4♥ bid should be sufficient to wake me up even without the alert.  Behind screens I wouldn't know about the alert.

(Partner decided to play the 4-4 fit in order to get a pitch on the 5th ♠.)

If I show no signs of distress when partner alerts, may I try to find a playable spot?

Since 3NT could be passed, I would have mild slam interest at most.  With a great fitting hand, partner could cue bid a minor, then we may find a slam.  After a 4♥ or 4♠ bid, could the 3NT bidder have slam interest?

If the bidding went 2NT-3NT-4♥-4NT-P, how should the director rule?