In US law there is a huge difference between a natural-born citizen and a naturalized citizen. Natural-born citizens are presumed to be loyal to the US. People wishing to become citizens are not, therefore they must renounce all foreign titles in order to do so.
Harry would have to renounce his titles to get citizenship. At that moment Meghan would automatically lose her titles because they are courtesy only based on her husband's titles. If his are gone, hers are automatically gone. She could not keep her title like Diana and Fergie did because her husband would no longer be titled. A divorced woman can keep her title in the approved form for a divorced woman unless she remarries, so Diana went from HRH The Princess of Wales to Diana, Princess of Wales. Likewise Fergie went from HRH The Duchess of York to Sarah, Duchess of York.
Archie could retain his titles if his birth in the UK is registered by Meghan, a natural-born citizen, with the US Embassy in London before Archie's 18th birthday. That registration, which has probably already been done, makes him a natural-born US citizen by right of Meghan's citizenship. Archie, as a natural-born citizen can then keep his title, which comes from his grandfather the King, not his father. Lili can also retain her title given by the King because she is a natural-born American citizen.