In Britain the possible variations in pronunciation are not ‘slight’ in the least, believe me. For example, in Geordie (the dialect of Newcastle, and more loosely the area of North East England around it, although the natives contest this), our is pronounced as a (non-rhotic) wor. Go 140 miles northwest to Glasgow, and in the local speech it’s a (rhotic) oor.
You ask: what do teachers do about it? I don’t think they do anything, tbh. They don’t need to. These variant pronunciations of the word are not random, but inherent in the dialect and accent in which they occur. In Scots, for example, words that in British RP are pronounced with an ow sound are regularly pronounced with an oo; e.g. flower becomes flooer (and is so spelt in written Scots). If a Scots speaker takes lessons in RP, the teacher will address this consistent difference, by making him/her repeat ‘How now, brown cow’ or some such method, and once the pupil has mastered this the pronunciation of our will fall naturally into line whenever he or she speaks in RP. It would be absurd to pick out our for special treatment. If a schoolchild in Newcastle says ‘Coom doon the toon, wor Jonty!’, no teacher trying to impose RP is going to say ‘No, no, Billy - it should be ‘Coom doon the toon, our Jonty’.