US Platinum cardholder here... and can report that I've been waiting 2.5 weeks so far. Took almost two weeks for Amex to process the enrollment (not sure why it took that long, it seems like they are usually processed a lot quicker) and am now waiting for Hilton to process the data received from Amex to upgrade the account.
Unbelievably I submitted a request for Hilton Gold on Tuesday and the next I got an email from Hilton saying I had been upgraded to Gold! It's never been so fast before.