It's not up to Sky or NOW to warn people when rights expire. On NOW they do have expiry dates available when its within x number of days.
That said, Barry is done by HBO and as part of the Sky deal we should expect to see it return on and off until 2025.
Never watched Barry myself.
Where the latest Season sometimes only gets a small amount of time on it's first run.
Usually on most NOW playback devices there is a warning when TV show episodes or a Season is leaving soon.
No idea about Barry where I have never watched it.
I also double checked this useful third party website link below and Barry Season 4 isn't listed anymore which again suggests to me Season 4 licence rights have finished.
Perhaps drop NOW an email using the address in the screenshot below for more information and ask them if they know when Season 4 will be returning (if known).