Jerryo1578, you're certain that you're logging into your Roku account at https://my.roku.com using the email address that's displayed on your Roku device (that you're using to access Brit Box) under Settings->System->About? Plenty of people have had charges that don't show up in their accounts and then it turns out that they have multiple Roku accounts, but if you can access it then that must be the Roku account with the subscription.
I don't know if it matters, but I assume you're accessing Brit Box through their app that just has "brit box" on the Home Screen icon and not "britbox on RokuChannel". If you have both icons on your Home Screen, does either offer you the option to "Manage subscription" when you highlight them and press * on your remote?
I have no doubt that the Roku subscription handling is screwed up. I have an app that gives me the "Manage subscription" option when I search for it, but not if I use the * on the Home Screen. Not only don't I have a subscription, but I know for a fact that the app doesn't offer any kind of subscription since it's my app - which makes me think that maybe you should try searching for "brit box" and see what comes up when you select one of the options that appears. I get the "Open app" and "Manage subscription" options on my app.