Just get some cold gun bluing, remove all rust, wash in detergent, rinse, degrease with alcohol. Use rubber gloves and do NOT touch the axe head while cleaning it. Skin oil will thwart what you are trying to do. When It is totally clean and degreased. Blue it just like a gun.
After it is blued, coat it with Ballistol (this is available in the U.K.) Ballistol is amazing stuff, great for carbon knives by the way, all natural ingredients. Won't harm humans if ingested. Its main feature is that it is oily, and when water gets on it, it emulsifies the water. Otherwise the water lifts the oil and comes in contact with the underlying metal.
Some American axes come, or at least they used to, come from the factory blued.
Blued metal is not rust proof, but it is resistant to rusting if coated. You will have a lot more attractive axe than a painted one. Gun Blue is cheap, and a little goes a long way.
