Nothing! Because they already use that power, just not in public. They blatantly exercise their powers to extort judges/politicians/media to their benefit.

They'd be deposed, obviously. Thing is, the very act of deposing them would be an extralegal action, as all laws in the UK are established under the crowns authority. Legally speaking it would be the equivalent of a non-violent coup. Hence ceremonial by convention(de facto), rather than by law(de jure).

There would definitely be a huge backlash. But also a very vocal minority who would probably support them. The end result would be some long overdue changes to our constitutional setup, but given the current state of UK politics, this would be delivered at the end of a highly polarising and disruptive period (I'm thinking Brexit vote level of rancour).

Would be better if we just nipped this in the bud now. Remove the monarchy's constitutional power, and make them fund themselves with their huge existing wealth. They can keep the titles and ceremonies for all I care, just pay for it themselves

Every new law passed is signed off by the monarch, a new PM still has to ask royal permission to form a government etc. In practice, these are purely formalities and are treated as such, but still legally there