What happens if I buy the Deluxe edition after playing via EA Play?

NoSundae​  If you buy a standalone copy of a game, you'll keep that when your subscription expires.  There is sometimes an issue with the EA App thinking you don't have access anymore, but at worst, you'd need to contact EA customer support to get that sorted out.  It's mostly not a problem though.

Saves should be intact regardless.

I currently purchased the standard edition, but through EA Play I have access to the deluxe edition. Once my subscription ends, will I still be able to play the game, or should I purchase the deluxe edition upgrade?

NoSundae​  If you don't buy the deluxe edition, you'll still be able to play the standard edition once your subscription ends.  Whether your saves will work depends on the game, although if the deluxe items are cosmetic, that shouldn't be a problem, in theory.  I have seen reports of saves not being compatible after the "downgrade" though.  If you want me to do some googling, let me know what game this is.

As an example, Sims 4 saves will still be playable with the deluxe content removed; you'd get a "missing items" error message, that's all.  Sims 3 saves would be unplayable or at least corrupt beyond repair if you didn't also own the Late Night expansion that comes free with an EA Play subscription.