You are overthinking this and completely off base, in a good way, worry less. There's only 2 things you need to avoid, "the man" telling you to take it out at work, and getting it caught on a lift as it swings around, THAT will cause problems
A piercing in your nose is going to react with temperature the same way it does in your ears, and the foreign object has no bearing on frostbite. If there is frostbite danger, there's frostbite danger, that piece of metal doesn't make the air around it colder. TBH you are probably not even going to notice it at all.
Seriously. I've worn zero's in my ears, including a 10g in cartilage, as well as had my nose pierced for a short while. Congrats on your new holes, tats are more fun tho
Making sure to put sunblock on the tip of your smeller is going to go alot further toward your health.
Disclaimer: yes I have worked outside (roofing, 6 years Chicago/Omaha) in Midwest winters with a various assortment of gear in my face, so I have a relevant point of reference.