There used to be the situation that more offers were available on the website than on the mobile app, inferring that there may have been a limit on the number visible in the app, which if true would lead to the impression that activating offers caused more to appear. However, of late, I believe all offers are visible on both at the same time.
I have not used the BP offer, however, if you contact AmEx all they will do is refer you to the Terms and Conditions, that sometimes say up to 90 days.
Longest I have ever waited for an offer credit to appear is about 15 days.
My latest AmEx offer credit (ATG) appeared in feed inside three days.
Amex statement offers in general have become something of a minefield; they used to post so reliably - notification immediately on authorisation then credit after the transaction posted. Seems totally random now - occasionally they work straight away, more often they show up a month or two later… very annoying when you’re waiting to close the card down (although I suppose that wouldn’t be a customer complaint they’d prioritise particularly highly…!).
The PayPal offer is a load of crap. Trying to pay for something through PayPal and Amex and my Amex card won’t show at the check out, but shows in my saved payments methods