The ship is British because it is on the British shipping registered. That means it has to obey all British marine laws in relation to safety, crewing, pay etc and also pays taxation in the UK.
Companies register ships in countries like Liberia and Panama because their regulations are far fewer, so ships considered to badly maintained to operate under British law can still plie the seas registered in Panama.
It is well explained here en.wikipedia.org/wiki/Ship_registration
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The crew isn't British. I don't think there is anything very British about it.