My reading of the palace source's statement as quoted in the Times is that Prince Harry's status - namely, having an HRH but not being permitted to use it - extends to his children now that they have become Prince and Princess like their father.
However, they will not be HRHs. A palace source said: “The use of the style HRH would come through their father and the Duke of Sussex’s HRH is in abeyance.”
Sources close to the Sussexes pointed out that Princesses Beatrice and Eugenie still have their HRHs even though their parents no longer have theirs. The palace argues that that is because they already had their HRHs when their parents lost theirs.
https://www.thetimes.co.uk/article/princess-lilibet-christening-harry-meghan-2023-nppf8tc7v
The palace's argument distinguishing Beatrice and Eugenie from Archie and Lilibet seems natural. It is the usual way of European monarchies that the removal of a right from an individual royal extends to future descendants but not existing descendants. For example, under the
Succession to the Crown Act 2013, if Prince Harry were hypothetically divorced or widowed and then remarried without the permission of the King, the prince would be excluded from the throne, and so would any future children from the unapproved second marriage, but Archie and Lilibet would remain in line. Other European monarchies have laws which function similarly.