No it doesn't. Apple Pay can notify you if you've connected the Amex there, but the notification doesn't come from Amex app, at least on iOS.

Edit: It seems the functionality is there, but it doesn't work for Contactless, which is the majority of transactions I've done.

I have both an Amex and a Monzo account, and the Monzo account is far more reliable than the Amex when it comes to instant notification. Yes, both have this feature, but I've found that only Monzo does it 100% reliably.