I think I've found the solution
The reason you get the login 1610 error is because the email you used to sign up with now tv is the same as your Apple ID which your Apple TV will recognise as being linked with the store.
I use iCloud for my email address which allows you to create an alias account that is still linked to your regular Apple ID
I found the best way to login to Now TV on the Apple TV is:
Create an alias email address in your email provider of choice, I use iCloud so created an alias email address in mail preferences on iCloud.com
Then login into now tv and change your email to the new alias.
Then log in with the new alias on the now tv app in Apple TV.
This should then work.
Your original email is still valid and the alias just re routes any mail you get to the original email address.
Hope this helps. It worked for me without having to reset my Apple TV
I’m now encountering this issue now the Beta isn’t running anymore.
I’m not changing my email address (also an iCloud email address) because that’s a workaround and not a solution. The email worked fine with the Beta app.
Please sort this.
I'm experiencing this issue and having spent a solid hour trying to find a solution still nothing? Can Someone at support give an update on a permanent fix to this issue? I have Amazon Prime and Netflix and don't run into this issue with either of these applications