I presently believe there is zero chance of WWIII resulting from Russia's invasion of Ukraine and the war crimes that it commits every day. This is because Europe is nearly irrelevant and the USA is not interested in anything other than trading Ukrainian lives in return for depleting Russian materiel. When we consider that sanctions are ineffective (because they always are) and that NATO explicitly tells Ukraine not to strike targets inside Russia - thus making it impossible for Ukraine to defend itself against endless rockets and missiles aimed at civilian structures - it's clear that no one in the West is interested in Ukraine defeating Russia.
The sad fact is that Putin will likely achieve 80% of his aims in Ukraine eventually, because we in the West lack the willpower and the capacity to sustain aid to Ukraine and without our aid, Ukraine is defenseless.