World War III isn't going to happen because this incredibly clumsy and chaotic invasion is proving that Russia is pretty much a random middling country with nukes. Their regular army would be wiped out by a professional NATO army within a week. They are using military radios from the 90s and have problems supplying their own troops right outside their own borders, for god's sake. Poland alone would wipe them out. And if they go straight for the nukes, you don't really have a war, it's something else.
This war started on the (utterly delusional) belief that Ukraine was a pro-Russian state that had been kidnapped by a US-appointed pro-western dictator who was barely holding on to power with the help of the US, and that the people and the army would immediately switch sides, grateful to be liberated. Instead, Zelenskyy now has a 90% approval rating and they are (rightly, and thankfully) bouncing the EU into becoming a more integrated block and opening the doors to them. Russia is being humilliated with every passing day without capturing Kvyv (US intel expected it to fall within 96 hours, we're past hour 100), they've lost more troops and materiel than the US did during its entire invasion and occupation of Irak, and the surprising unity and strong Western sanctions are collapsing the Russian economy into North Korea but with oil. It's an unmitigated disaster. It's what happens when you get drunk on your own kool-aid.
There's actually Ukranian soldiers reporting that some of the Russian troops they caught were just kids (18-20) with no training and very poor equipment. It's so sad to see