“Unwritten constitution” is bullsh*t. A constitution must be written. Saying otherwise is taking the p*ss. It would be amusing, were it not so sad, when this phrase gets parroted by British journalists, politicians, and experts. Not one of these idiots has ever considered whether or not it stands up.
It means “Don’t worry. Privately-schooled poshies have your best interests at heart; they know they can’t simply do whatever they want; and that’s the way they like it.” But of course if it were actually said like that, it would be a case of “I’m not lying to you”.
The phrase means “Don’t have a revolution”. It means “tradition is democratic”. See what I mean about taking the p*ss.
It is not surprising that it was coined around the time that revolution was going on across the Channel in France.