Sorry that of course should have been "Good post Lizzie":

The Duchy of Lancaster:
The duchy is not the property of The Crown, but is instead the personal (inherited) property of the monarch and has been since 1399.

The Duchy of Cornwall:
There is a large body of evidence collated by the Attorney General to the Duchy of Cornwall in the 1858 Cornish Foreshore Case that showed Cornwall to be extra-territorial to England. For example on George III's accession on 25 October 1760 he surrendered 'Crown lands' held in England, Wales and Scotland to the state in return for an annual 'civil list' payment. The Duchy of Cornwall, i.e. Cornwall, was excluded from this legal arrangement, so today the Crown Estates has no jurisdiction in Cornwall.

High Sheriffs must by law swear an oath of allegiance to the sovereign, but the High Sheriff of Cornwall swears an oath of allegiance to the Duke. Nearly all laws passed for England and Wales today do not apply to the Duke or his territorial possession known as the Duchy of Cornwall and only two named individuals in the UK have Crown Immunity – the sovereign of the UK and the sovereign of Cornwall "in right of his Duchy of Cornwall".

Various laws, and the government's own legal department, state that the Duchy of Cornwall is co-terminus with the administrative area of Cornwall and to this day, the Queen is the owner of last resort of all land in England, Wales and Scotland, but not of Cornwall. These exemptions (which are applied on grounds that the Duchy is not part of England and Wales), reflect a constitutional situation that the Duke is able to protect by right to intervene in, and control the outcome of, any court case that might affect the Duchy's unique constitutional position.

Please note leveler:
"On George III's accession on 25 October 1760 he surrendered 'Crown lands' held in England, Wales and Scotland to the state in return for an annual 'civil list' payment."

If that ANNUAL 'civil list' payment is not paid to the King or Queen, because they have been replaced by a President, then the contract made on the 25th October 1760 is broken and Crown Lands revert to their rightful owners.

It would be a fascinating case, but one great thing about English Law, it is absolutely clear when it comes to property title and protection of property. My guess is the Queen or her descendants would win, worst case they'd come out of it with one hell of a settlement.