you cannot build an underground shelter under extreme heat.

Yup. That's why I said,

"but they would have had to start that project some time ago"

When the day comes do you think there will be room for everyone?

Depends when the project starts, how much money goes into it based on the nations priority. There are subway systems in some countries designed for this purpose that can hold entire cities. Not entire nations but enough that a nation or civilization could be restarted. It would be a very rough start.

If countries started putting a small percentage of their GDP into building underground cities a few decades ago they could probably save most people. They would have to store up massive amounts of freeze dried food and have water treatment facilities that can hold lakes of drinkable water.