You don't have to wonder, the spell it out on their website:

We are not a political organization. We don’t stand for, represent, or align ourselves with any political party.

Still, I think campaigning against specific policy would meet the average person's definition of political organisation.

Kinda like if the King would say climate change is bad, it’s political but it doesn’t really compromise his position as apolitical

Maybe when all the (major) parties agreed that climate change was real you could argue this, but now both Reform and the Tories are both doing climate change denial I don't think you can. Besides, the idea that the monarch is apolitical when it uses its privileged position within the British state to lobby for the monarch's personal interest against things like workers' rights and, ironically enough, climate laws is plainly absurd.