My theory is that BritBox's metadata for these shows routinely if not at least frequently doesn't allow for the TV+ channel subscription to play it. Note that this is purely theory as I have no idea what any metadata. actually looks like. Attempts to get this resolved via both Apple and BritBox have been similar to what's already been mentioned here -- neither understands the problem and neither takes any action to resolve it. Even when I mention I'm a TV+ subscriber to BritBox with an old, expired direct BritBox account that I don't use any more, the first thing BritBox always says is, I see your account expired.
At least for my experiences, the shows eventually start working, a few days later. Usually, so far.
I used to have a subscription directly from Acorn TV but they cut me off halfway through an annual subscription and refused to acknowledge the annual payment evidence provided. As such, they're kind of on my "do not touch" list now.