Unless the King issues new Letters Patent, they already have authority from the sovereign from the extant ones.
I doubt either will ever use HRH or Prince/ss, given earlier comments from their mother, but that's down to upbringing and expectation.
They had to change the LP for the Cambridges, and do so before any of them were born. Because otherwise had George been Georgina,, and next in line to the throne, she would still have been Lady Georgina Cambridge (or Mountbatten-Windsor). Whereas her younger brother would have been HRH Prince Carl - because he was eldest son of grandchild in direct line. And then little brother would have been Lord Louis. Obviously a nonsense, and I think the right call was made to give all siblings the same royal style. But it could have just been changed for the eldest to be HRH Prince/ss.
The changes were made before Harry had even met Meghan, so I don't think his choice of future bride was a consideration. Harry would have known this, and should have been able to explain it.
His DC could have been Lord and Lady (customary to use subsidiary title, but not a requirement)but they chose to eschew titles them, and do not seem to be taking up the new royal ones now. Which makes sense, and has Wessex precedent