WW3 will not even start in case of one isolated nuclear explosion (Russia bombing one US city), there are limited response scenarios prepared for that.
On the production level, conventional war between Nato and Russia happens, and the BBC presenter is oh so calm and composed, but I supposed you can't ask much from a low-budget documentary (if I can call it that).
0. See WWI.
Also the war did not "start over an assassination". It's much more complex than that, and the assassination was just the pretext.
See for example the lawful (but provocative) downing of a russian plane by Turkey. The retaliation from Russia was basically just words and some economic sanctions.
I'm not sure if the leader of US or Russia can order a first strike without inbound missiles from the other side. Would the military execute a first strike knowing what the response of the other side would be?