You're in the UK, so you'll see a lot more people talking about the British Museum, than, for eg, the Louvre.
The British Museum is also unusual for having quite a lot less national/local stuff in it than any other national museum. The Louvre's collection is about 75% French/European. The British Museum is only 8% British stuff.
The history of the British Museum is also extremely tied to the British Empire in a way that, for example, the Louvre isn't. Sorry I keep comparing to the Louvre but it's the only other international museum I know much about.
The British Museum was explicitly set up to house the spoils of Empire, whereas the Louvre is a post-French Revolution invention. France did loot extensively also of course, but most of their stuff is housed in specialist museums rather than the big national headline museum. The way the British Museum is set up as the nation's greatest museum while almost exclusively being full of loot is quite jarring and attracts conversation.
The British Museum's culture itself is extremely conservative, especially compared to many others within the arts and heritage sectors. I mean, the chair is George Osborne... they don't discuss the pillaging and looting in any of their communications in the museum itself, which again is quite jarring. This also leads to other people having the conversation about them, rather than the institution being part of it, which is how a lot of other institutions are responding to this cultural shift.
The British Museum is forbidden by Parliament from giving away any of its objects, so it can't really have a woke awakening because it might get it into hot water legally. But it does mean that its sensibilities are not quite aligned with modern culture and people feel genuinely angry about not just the collections themselves but the way they are marketed and communicated.