We usually pop them on after mum has licked the lamb off, but we have put them on earlier especially if mum is slow to lick off or they look like they are loosing too much heat, then i dry off with hay or clean paper towel and stick the jacket on.
What's more important is getting the lamb on the teat for internal heat, that first feed is the lifesaver. They usually dry off a bit under the jacket, but can sweat a bit if too hot, ie in a shed.
If the lamb looks wet underneath then we pull off the jacket and dry off again.
We only usually leave them on for a few days if weather is bad or lamb a bit tiny. Sometimes mum will remove it for us anyway!