Although on a somewhat serious note:
The Canoptek wraiths/spiders/scarabs wake up first. Technically they have never been asleep but now they go from Maintenance mode to full on Wake up mode.
After this it gets a bit confusing and it starts to depend on circumstance. I get the impression the first legions to wake up do a recon sweep to check for Aliens and they try to evict them. If they are unable (or if a certain plucky Commisar pours a refinery worth of burning oil into the tombs) then the wake-up sequence is accelerated and with precedence to the militarily might of the tomb.
Imotek, the senior comander of his world, is mentioned as waking up to a mostly functioning tombworld so apprently the command structure is not the most important thing to emerge. Maybe the Immortals just wing it or maybe some junior lords set things up while the Phaeron hammers his snooze button.

If no squatters are detected I'm not entirely sure what happens but I'd imagine the whole sequence is pretty much the same only slower and more deliberate.