Rick rightly points up the issue of drying time when wooden coops are treated with any sort of preservative. In direct, hot summer sunshine, effective wood preservatives (those continuing spirit-based ingredients, or traditional creosote) take at least 48 hours or more to dry and for the fumes to disperse enough for the chickens to return. In normal cool UK conditions, and especially in freezing weather, you're likely to be stuck for days, or even weeks in winter, consequently you need a second coop for the hens in the meantime.
I think painting the inside a coop is unnecessary in any case. It would do nothing to stop redmite, best to stick to diatomaceous earth slurry painted into the cracks and joins. The walls just don't get dirty if theres enough perch room for the hens to roost, and redmite are not found on walls unless you have a mega-infestation.
Or get a plastic one, of course, and save yourself years of problems.