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I would look at Azamara - small ships, no formal nights, no facilites for children - she is doing world cruise in 201.
So you want a world cruise out of the UK - your choices are limited to Cunard, P&O or Saga then - all of these have formal nights, so I think you are out of luck. Azamara is a good alternative but her world cruise is out of LA and ends in NYC. Perhaps you had better start your own cruise line as you have such specific requirements.