In gentle, low speed use, I don't think there is a significant difference between a One and a Cooper for economy. The One is effectively the same engine as the Cooper but with its top end power limited - as you aren't going to get near that range around tow, I would expect them to be the same.
But I haven't owned both a One and a Cooper, so I can't say for definite.
What does make a significant difference is whether it is a 1st or 2nd generation car, with the 2nd generation ones being significantly more economical. The changeover was in 2006 (except for convertibles). They will probably have lower tax for the same reason.