NATO as in, full-scale involvement of the U.S. Military and assorted EU Military forces? Tough to say honestly. I would obviously root for the NATO forces, as well, I fully expect a potential conflict with Russia during my time in the service. And I am hoping to go Armor branch, specifically looking for either Tank, Cavalry, or Mounted/Mechanized Infantry command. Those three will be the bulk of any territory pushes and will be require a massive number of units.
and oil, thankfully, shouldn't be much of a worry. While yes, they have vast amounts of oil available. If melting in the arctic occurs, I envision the Navies will come into their own large-scale battles over the Arctic Ocean floor where we have been told unimaginable quantities of oil exist beneath the rock. This will be a precious commodity during the war so that the war can be waged effectively and to prevent transportation crises across the world due to the military consumption. Russia has already publicly pointed out that they want to claim it for themselves. This will almost certainly mean it would be a hotly contested territory during any US/Russian conflict. I've heard that Canada also would like to own the property.
Oh, and even if it weren't contested or Russia claimed it, the NATO war effort will be supported quite likely by quite a large percentage of the oil-producing nations. One, a lot of them in Asia have unfavorable opinions of Russia, due to history with the USSR. I am sure world opinion, except for a few nations nearby, such as N.K. and China, will heavily favor the NATO effort.
All of the above is at least what I hope will happen in the case of NATO/Russian conflict, because I don't want my command to be effectively neutered by lack of fuel.