The Blackheath Society used to be run by an old misery who appears to have objected to pretty much all forms of human progress – including the tea hut. There’s a great archive of old photos on their website, including some of the tea hut, where you can tell that they came from him going up with his camera in the 1970s onwards to take photos of it looking grotty.
https://charltonlife.vanillacommunity.com/discussion/87905/blackheath-tea-hut/p5