As of today, Archie and Lilibet are legally HRH Prince Archie of Sussex and HRH Princess Lilibet of Sussex. It is unknown whether they will actually use the title or follow James and Louise's precedent, but unless the King rescinds it, the title of Prince/Princess is theirs by right as children of a son of a Sovereign of the United Kingdom.
As for the Cambridge children, as of now, they are styled HRH Prince/Princess [xxx] of Cornwall and Cambridge until their father is created Prince of Wales, which is not automatic and depends on the King's pleasure.
And it looks like that, as it had been previously announced, the King's wife will use a rather unusual style of HM The Queen Consort, rather than the usual HM The Queen.