Are you using a VPN either on your TV or Router?
If so switch off the VPN.
If you have any other NOW playback devices are you having the same sign in issue problem with the App.
Maybe also try a fresh App download and unplug the power to the TV for about 2 minutes before attempting to sign in on the NOW App.
If you have mobile data perhaps try a wifi hotspot with your phone just to see if it let's you sign in on your TV.