Unless you have a mite problem, there's no reason to treat a bird. I'm not familiar with mites that you may have there, but mites are usually active at night & if you place a white sheet of paper in a cage, at first light you should find them on the white paper.
Not sure about coconut oil as a mite killer either, but with a bird, you have several layers of feathers to get through before you reach their skin, so you would almost have to entirely dip the bird into a solution for it to reach the skin and any oil residue will make it look much like a seabird that has fallen into an oil slick.
For bird feathers, think fingernails or your hair.....the nail-bed & hair roots are still growing, but once they are outside of your body, they are no longer alive and would not benefit from any oil solution.
Doesn't the UK average cold stay around -2C? That is sweater or jacket weather on this side of the big pond. When it gets below -10C we start putting on coats.....I had the experience of spending a week around -50C one time.....that was cold.
Unless your feathered friends are allowed to run with chickens or wild birds, you should never have to medicate them except on advice of a veterinarian and just because something is sold in a pet shop, it is not always good for your pet.....also, you want to be careful about mixing anything in a bird's water, because many birds will refuse to drink it due to the different taste and can actually cause themselves to dehydrate.