"But what's going to happen? Will IANA stop giving out numbers and say, "sorry, nothing we can do. No more numbers."
Yes. Quite possibly in less than a month.
When we're down to 5 /8's they're distributed to the RIRs automatically and IANA shuts down its ipv4 operations forever.
The RIRs then have until their v4 pool runs dry - they won't get any more - which may be quite quick for some (like apnic) and slower for others.
After that it's down to what ISPs have - they'll probably ramp up their prices for v4 an instigate such horrors as carrier grade NAT, just to stretch it out, but it's just postponing the inevitable.
And yeah, potentially this is bigger than y2k and the great unwashe don't even know it's happening - even though it *will* affect them.