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Lyon is not the most vegan friendly place on earth, but found a place called Le Jardin Interieur - it was very good, if pricey and with a very small menu, which oddly always has one meat dish. It's also quite a walk from the centre. You may be better off just finding a pasta place in town which has a vegan option - found a few of those.
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just in case you've not seen it, this thing can be quite handy -> https://www.facebook.com/groups/211687469564055/
Well, with London and the Shacklewell Arms you won't be a million miles away from Hackney Central. Loads of places, including Black Cat Café (the closest and longest-established (in various forms) in Clarence Road) Temple of Seitan, Vegan Sutton & Sons in Graham Road (fish and chips), Cook Daily in London Fields, Rehab at Well Street/Morning Lane, Dark Arts in Ponsford Street (I haven't been there yet, but @mrs_socks has and said it was very good), the Spread Eagle in Homerton High Street (Club Mexicana), and some other recent openings, but that should do for starters. :)
In Brixton, you have the 'Fauxmagerie' in Brixton Market, Ms Cupcake, Café Van Gogh further up Brixton Road--and of course you need to go to Brixton Cycles. There are probably many more things, but I'm not local to there.
I don't know about New Cross, sorry.
Are the band vegan?