To be clear, NATO is NOT deliberately going to war over Crimea, or any other part of Ukraine. The only way that might happen is with Russian nuclear first-use against Ukraine. Even in that scenario, there is a high chance of both sides backing down rather than taking things to the next level, depending on who exactly is calling the shots in D.C. and Moscow.
A NATO-Russia conflict on anything more than a border skirmish level would be the most destructive conflict of the 21st Century. We're talking about Russian troops and soldiers from NATO countries, with both sides equipped with 4th and 5th generation fighters, strategic bombers, cruise missiles, third generation MBTs like the T-90S, M1A2-SEP, Challenger-2 etc, APCs & IFVs such as the BMP-2, Warrior, M2A3, Howitzers like the M109, MLRS, and toys like thermobaric weapons, clashing not just in Ukraine but in the Baltic States and the Baltic Approaches, Belarus/Poland (depending on the course of the war), the Atlantic, the Black Sea & the Med and possibly in the Pacific. And the fighting wouldn't just be contained to the front lines. In any serious conflict Russia would strike NATO harbours & airfields in the UK, Low Countries, Germany etc and NATO would be hitting Russian road and railway nodes on Russian soil.
You're looking at 100,000 + dead on NATO's side, with countless Russian casualties. Civilian casualties would be higher than anyone would like to think about. That's assuming the nuclear genie remains in the bottle.
However, Moscow was definitively prepared to utilize Russia's nuclear arsenal if NATO attempted to interfere in the annexation of Crimea. Should NATO have decided to act more strongly, with advisors and weapons being sent to Ukraine, ships in the Black Sea, CAPs on behalf of the Ukrainian Air Force, amongst other measures, things have a very small, but very real, potential to spiral out of control after a series of miscalculations or mishaps. The only way a war happens if both sides react strongly to provocations by the other, and this leading to a shooting war; NATO deliberately going to war with Russia is not realistic; blundering into conflict is far more likely.