I don't know enough about who they're polling to know for sure, but I suspect if you polled Brits randomly, you'd probably end up at least at something like a 1:5 ratio minimum, not 1:20. If only 5% of the population was on their side, they'd have to step up their game from disenfranchisement to outright disappearances. 20% is probably just enough to control a liberal democracy through mostly non-violent, mostly-institutional means, as long as you keep them politically active, like by making them fear their holy mission might fail.

News media and its offshoots aren't usually engaging with dissidents directly. The MSM is there to control the idiot masses through false consensus, and distribute the political orthodoxy to its followers. Through that lens, it makes sense for BBC to try to scare people by presenting a widespread "threat" to the orthodoxy.