Perhaps I'm not explaining it well enough or my choice of words is poor. I can see there's a certain amount of conflagration of issues through the thread.

I'm saying that I don't see any reason why an officer would deny entry based on the likely evidence that could be presented, whether that's TWOV or with a transit visa, hence my reference to the UK having the ability to refuse anyone, irrespective of whether they have a visa or not. I was trying to make the point that it's not very likely to happen with either route.

But, if the OP is concerned about being refused either by the airline or by the Border Officer, they could choose to apply for a transit visa given we know they meet the criteria from their own posts (I think!). That would be a further layer of security.