My experience is that there's a manual check, and it's sometimes done almost immediately, sometimes it takes a bit longer, it depends on how complicated EE have made the plan. You won't need to do anything, you should get an email saying it's been done and see it reflect shortly after in MyEE.
On that basis, when does the new plan become active? Do I have to switch the phone off for the old sim to reboot, so to speak?
Not necessarily, all you're changing is your payment plan - there's no need to make any changes to your SIM at all.
You'll probably get a few text messages sent out to you to confirm the change, but otherwise your SIM has no need to even know your plan has changed - it's pretty well irrelevant.
