Yes. I think it's clearly established you need an Act to remove a Peerage. Whether you need one to remove the title of Prince seems more contested. In 1917 Parliament was first asked to pass the Titles Deprivation Act.

A good briefing here, though doesn't resolve the issue directly. commonslibrary.parliament.uk/research-bri...

Though genuinely this seems to be a contested point. On the one hand, an Act of Parliament was sought (and therefore thought required?) to deprive Ernest Augustus of the title Prince in 1917. On the other hand, the title is conferred by Letters Patent and so the royal prerogative.

The House of Commons library briefing seems unclear on this: commonslibrary.parliament.uk/research-bri...

Yes, I think that's the correct reading. You need an Act of Parliament legally to deprive someone of a Peerage (and the title of Prince).

Precedent suggests the government will need to lay a Bill before Parliament to turn Prince Andrew, Duke of York into Mr Andrew Mountbatten-Windsor as a matter of law. They should do so.

Renouncing peerage doesn't seem to be the same as being deprive of a peerage, though.

We're undoubtedly in the penumbra of the cardinal convention here. Evans itself recognises exceptions to it. Perhaps the regulation of royal peerages is one.

But presenting this as a quasi-private decision of the monarch seems questionable from the perspective of constitutional convention.