Hi there,
This is indeed an interesting question which is probably relevant for quite a lot of users, so I am happy to provide my perspective on it:
To my knowledge and the experiences of coachees of mine, BCG does not conduct a written case in London.
In case you want a more detailed discussion on how to best prepare for the upcoming interviews with BCG (and maybe other companies as well) in London, please feel free to contact me directly.
I hope this helps,
Hagen