I think any model adopted here would have to reflect our history. I don't think it would be helpful to look towards other countries because we have no history of anything other than monarchy (not really, anyway, and certainly not in recent times).
I think I'd keep the Queen (or her equivalent). When she dies, her heir should succeed her. She would be able to veto some things, call Parliament back and declare war. A bit like now, really. But she should not interfere in the business of everyday democracy. That should be up to elected politicians.
Her children should not be royal. Her husband should be simply that-her husband. She would be allocated one home which should also serve as her office. The homes she currently has should be managed and owned by the state, as should be the Duchy lands.
We should not be subjects, but citizens. The Queen should be tested and declared "fit for purpose" once, say, every five years.
She should be given an income on which she should live comfortably. If she is unhappy with this arrangement, she has the entitlement to step down.
We keep the Queen as figurehead, but with greatly reduced riches.