It might take 24 hours for the App to be pushed out to customers NowTV branded devices (so be patient).
You could try a system update in the menu settings just to see if the App is idle there waiting to go into the Apps Store section on your box / stick.
Just one word of caution the old white nowtv boxes will not be receiving the Disney + App from what i have heard and currently it's only being rolled out to UK customers where Republic of Ireland customers will have to wait a little longer according to NowTV over on their Twitter feed.
I've got the latest NowTV stick and there's no Disney+. Tried to update through the system but it said it's already updated. So I guess will be trying again tomorrow. I have signed up to Disney+ and don't wanna watch via phone. Thanks for the info anyway.