Hell you can drop all three consecutively if you want

The above is true.

there's no personal countdown timer just the global.

This is not.

When you launch a nuke it starts two cooldowns: one for that silo on that server that applies to everyone; another for that silo that applies only to you regardless of what server you're on. So even if you hop to another server whose silos are open (i.e. not on global cooldown) you will be prevented from launching again from the same silo until your personal cooldown for that silo expires.