what happens if as a matter of principle she refuses to sign a bill?
The monarchy ends.
It probably depends on circumstances. It it is a bill that just about every sensible person thinks should be signed, then the monarchy ends. If it is a bill that just about every sensible person thinks should be vetoed, then the monarchy is strengthened.
it would nearly have been worthwhile to keep pre-decimal currency on those grounds, just so one could explain to American tourists how many groats make a florin.
As long as you can still bowl a maiden over, all is not lost.