Translation:There are no police boxes anywhere.
This was a little confusing to me because of the double-negative. どこにも means "nowhere" while ありません means "doesn't exist." Therefore, one might read the sentence as "nowhere the police box doesn't exist" if they're trying to work it out which, in English, means one does exist, but everywhere. I suppose どこにも would never be used with あります. Is that correct?
It's not a double negative. どこにも only means "nowhere" because/if it's combined with a negation, like ありません or ～ない, which in English translations often turns into "anywhere". E.g. どこにも見ません: "I don't see it anywhere".
It can also be used with あります or other positives, in which case it means "everywhere". E.g. 危険（きけん）はどこにもあります: "there is danger everywhere".
Is "police booth" incorrect? A box is for storage, a booth is for people to work in
Not the same thing. It just seems equivaent because of a translation quirk.