In England today, "put your coat on" and "put on your coat" are in free variation. But was there an original dialectal difference in the placement of the preposition, and if so, which areas said "put... on" (and take....out and all other similar examples) and which areas said "put on X"?
Maybe you are mistakenly assuming that with these two usages there is a difference in dialectal area or historical period. Two things to bear in mind: put on is a phrasal verb, and phrasal verbs can be inseparable, optionally separable, or obligatory separated. It so happens that put on, with the meaning don (other meanings exist), is an optionally separable phrasal verb. To my knowledge, it has always(edit) and everywhere been that. Thus, it is not that some related group of people says it one way and other groups the other, but that everybody has individual preference.
Also, the context plays a very important part:
The context for the first sentence might be a child bounding out the door on a first snowy day of the year. It's a taken that on such a day, something needs to be put on, and so only coat is stressed. If needed to be radically shortened, the first sentence could be: COAT!
The second one stresses both the need to don something and that that something is one's coat. It cannot be shortened. The context of a) might be a boyfriend going out to the store on a cold September evening after he has gotten used to dressing lightly during summer. That now something needs to be put on, is not taken. Thus, both the predicate and the object are stressed. (With the predicate, only on is stressed, due to phonology; on has only one syllable.) The context of b) might be a mother angry with the kid who keeps going outside lightly dressed despite catching cold every time he does. PUT on your COAT! Here too both the predicate and the object are stressed, but this time the predicate more so, because the donning (any extra clothes) is the source of the mother's vexation, whereas the coat then is more of a given. (With the predicate here, put is stressed, not on, because a wrong meaning would be conveyed were on the most stressed word in the sentence.)
Well, my "always," it turns out, is not quite that old.