Re: "milled somewhere that doesn't also handle wheat"
Wheat-infected oatmeal is not typical in Scotland, nor I assume in the rest of the UK. It mostly seems to be common in the US, where one very seldom finds natural, unadulterated food products, since nearly everything over there is heavily processed, genetically engineered and saturated with chemicals. I believe the most common contaminant is an abomination called HFCS - banned in the EU and elsewhere, a synthetic and sickly-sweet syrup made from genetically engineered corn, which is added to pretty much everything, including supposedly savoury dishes.
Over here I think you'd need to buy "instant oats" type products to get oatmeal that contained anything other than pure oats, and generally speaking such things are never used as ingredients for other dishes, such as haggis or black pudding. Plus no true Scotsman would ever ... etc.