Same here. After 10.3 all my devices were displaying "We can't show your Up next list right now. Try again later".
This morning, I updated my iPhone and iPad to 10.3.1 and the feature started working again for a bit.
Side note: I didn't reset my TV history as it is suggested by the solution post in this thread.
But this afternoon, it stopped working again! This kind of indicates that Apple is probably working on this feature on the server side. It could be an iCloud sync issue that only happens under specific circumstances which stops this service on an account basis. Also, I noticed the "Mark episode watched" appeared without a firmware update (on AppleTV 10.1.1)...which adds to the possibility that this was done on the server side.
At last, I checked the iCloud System status page to see if there was some indication of a temporary issue but there is no section for the TV App "Watch Now/Up Next" feature.
I'm surprised that there isn't too much buzz about this issue...which kind of indicates that it isn't a feature as popular as most of the people in this thread would like, including me. Or only a small portion of accounts are affected. Hopefully Apple will find and fix the issue soon!