As to why "queue" is spelled "queue" and not "cue", one can ask why "quay" is pronounced "key".

The British seem to have a certain fascinating eccentricity about such things [perhaps because they think they invented the language]. Americans are less given to these embroideries. For them, a "Quay" is a "Key", not a "Quay", and they will not stand for "queues".

Spelling it 'quay' is, indeed, a pain in the bum. And it's all the fault of pretentious nitwits who fancied themselves as sophisticated.

'Quay' used to be spelt 'key' (ME 'key' from OFr 'kay'). But the spelling was changed under the influence of modern French 'quai'. (Concise Oxford Dictionary, 10th ed)

Don't know why 'queue' is spelt like that, but it's from the French again.