I asked about the free evenings for museums that normally charge admission, because I haven't seen any other evidence that these actually exist.
Your reply then seems to only be about museums which are free anyway, so that doesn't really help. (I'm guessing National Portrait Gallery and V&A?).

(It's less relevant to this forum - but I'm also not aware of any museums being open 'all night' - apart from the 'sleepovers' they do. Many certainly do have 'late nights' - but nothing beyond about 10pm, as far as I know. I also think that you were possibly misled about the London Transport Museum being fully booked, I don't think they normally even take advance bookings except for groups.).