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On TFL roads Khan has final say, On borough roads Westminster would have final say. But Westminster's provision for cyclists is so shit, cf Baker St, that would be no great loss.
Khan should do what he can on TFL roads, show people how well segregated tracks work, then let political pressure work on the boroughs. His policy of "zero conflict" will get nowhere and has got nowhere.
Khan could have told Westminster to piss off but then they'd block any plans thru the borough I guess.