It’s not legally possible to move an airline to another country. Flights are granted as part of international treaties and Cathay has to remain a HK airline.
Actually, it's much easier than it looks.
CX has to obtain an AOC in -for example- Singapore ("Cathay Pacific Singapore", you name it) and subsequently can fly from there, as a permanent base. Requires a lot of paperwork, which will be a little different from the Hong Kong paperwork, but certainly doable. Airplanes may need to be registered in Singapore, though maybe just "wet-lease" from Cathay Pacific Hong Kong could be possible.
Of course, all the landing rights abroad will be tight to the Singaporean treaties, though these do exist, etc. And, given the under utilization of most airports around the world, the airport landing rights will be available (for free or a minimal fee).
Think about the AirAsia, JetStar, etc, subsidiaries. Several EU airlines do have a similar structure, think about Ryanair.
Of course, SQ won't like it to get a competitor "in-house", though it highly depends on what the Singaporean government thinks of that. They may like to see the SQ pricing go down to .....