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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?
Also, I'm off driving a band on tour starting next week, if anyone has any reccomendations for places in any of these cities let me know! (Excluding Paris and Berlin, I'm not with them for those dates).