Can a Pilot hold multiple foreign licenses and practice the privileges of it in each country? Is it possible for a CASA CPL holder to convert to both an FAA and CAAUK license and hold all three at the same time?

Would the pilot then be qualified to exercise the privileges of the CPL in Australia (with a CASA registered aircraft), USA (With an FAA registered aircraft), and the UK (With a CAAUK registered aircraft)?
Yes, a pilot can hold multiple licences from different authorities, but it's not about operating in a country. A license from an authority applies to aircraft registered with that authority, wherever that aircraft is in the world. 

An FAA license lets you operate an N registered aircraft anywhere in the world, it does not let you operate a foreign registered aircraft on American soil unless that foreign authority gives permission to do so.