You're probably out of luck if you can't access it through "The Roku Channel" app on your Fire stick, . Subscriptions through third parties like Roku often come with restrictions like the subscription can only be used on certain devices. That's just one reason why most here would recommend against subscribing to anything through a third party like Roku, Amazon, Apple, etc.
Is there any chance you can log into the Britbox website using the email address associated with your Roku account and their "forgot password" option? If so and you can establish a password, you may then be able to use that on your Fire stick.
I turned on a Brit Box subscription using my Fire TV television, via stand-alone Brit Box app, even though I was charged by Amazon I could use the email and password for the Brit Box app on all my Roku as well. That wouldn't be the case if I purchased the subscription in the Prime app, even though both are paid for via Amazon. The Same would go if used stand-along Brit Box app on a Roku, I would still be able to sign into Amazon device. But that wouldn't be the case if turned on the app via Roku Ch.
