Would this be a "bloody nose" as in his nose is bleeding?
No, it means that he has blood on his nose.
A bloody nose would be næseblod.
Vagten har næseblod.
But what is he only allergic to plants and therefore his nose is bleeding, would it be a bloody allergy nose or allergy bloody nose?
The former if it bleeds on the outside, the latter if the blood comes from the inside. :)