It will happen - both the main parties (Conservatives and Labour) have said they will abide by the vote. If they didn't they'd get crushed in the next elections - this is the beauty of Parliamentary democracy.
I know Greece caved - but that is because they are a poor country that is enmeshed in the euro. The UK is the 5th largest economy in the world and controls it's own currency.
Also, haven't you heard? Boris Johnson is out of the race to be PM (he got done in by some spectacularly ruthless shenanigans on Thursday).