Also, it would be so helpful if I could take it on the Lizzy Line from Woolwich
I think you can? Text here is confusing but look at the map and key https://tfl.gov.uk/modes/cycling/cycles-on-public-transport
Huh. D'you know, I searched and searched for info on whether you could, and all I found was a very old document that said "no bikes on sections that go underground"
That seems to suggest that I should be fine then, thank you for finding it!
@TooTallTim started
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I think you can? Text here is confusing but look at the map and key https://tfl.gov.uk/modes/cycling/cycles-on-public-transport