Etymology of 'rime' and 'unrime', meaning 'to put on/takeoff outdoor clothing'

These terms were in use when I was a boy in South London back in the 1930s/1940s. My grandmother would tell me to "Rime up well." or "Get well rimed up." when I was going to go outdoors on a cold day and when I came back I would say "I'm going to get unrimed", meaning 'take off and hang up my outdoor clothes'. I used the term today and had to explain it to my wife who said she'd never heard it. Now I can't see it in the O.E.D. Sixth Edition (2007) or anywhere else

I think I got this one. As Chappo suggests in the comments, rime is a coating. Moreover, while the dictionary definition he provides seems to indicate it is usable as a synonym for any kind of coating, I have only ever heard it used to describe the kind of frost that you get which thinly covers all the surfaces of a damp area once the temperature drops.

As it relates to your grandmother of South London, there's a lot to like about this etymology. First, rime is likely prevalent in South London due to the climate; it is liable to occur whenever a foggy day goes under freezing. Next, it is easy to see that she was asking you to get bundled up to protect against cold weather, which is indicated by the presence of rime on surfaces. Finally, it has a nice metaphorical component to it, wherein you protect yourself against the cold with your own coating, suggesting a sort of wordplay that, as an ugly American who has never been, I think of as rather prevalent in the British Isles.