Hopefully it's sunny and first thing I do is get my bridle and find a nice sunny spot out of the wind. By the time I halter my horse take to the barn, groom and saddle the bit has warmed up. But I also board at a small private place and don't have to worry about my bridle walking off.
I typically use my bare hands, or breathe on it! You can actually get fleece warmers for them that are $13ish, but last year I even used to use a old soft facecloth and took it off right before I use it, and it did the trick! Hope this helps!
We have several hot water sources in the barn so I usually wet a cloth with super hot water and wrap it around the bit as I'm putting the saddle on. By the time I'm ready for it the bit has come to a perfect temp. Sometimes I will run the hot water over the bit directly but the metal heats up within a couple of seconds so you have to let it cool before you bridle the horse.