Slow Horses season 3 won’t play. I’ve just subscribed, account info says I’m signed in but will only play episode 1 from season 1

Playing only S01E01 from select shows is a symptom of not being subscribed through the account with which you are signed in. These select first episodes are meant to be free to anyone (presumably as a sort of teaser).

For more information and assistance: Contact Apple for support and service - Apple Support

My knowledge of subscribing on Fire TV is limited, as that is new and I’m not sure how that is administrated.

Please contact the Apple Support staff. If they can, they will help you get the most out of your subscription, including during free trials.