This is not about whether WWIII will happen or not or who will start it, this is a simple "What if?" scenario.
The reason i am asking this is because the internet Didn't exist in the first two world wars so there is nothing to look back at to see what would happen.
Personally i think that countries would block their enemy's IPs and simply deny them access to their websites.
Maybe a lot of bandwidth would be taken up for intelligence sharing/communications causing people's connections to be slower.