What materials are suitable for a mini/model gas turbine engine, specifically the fan, compressor, combustion chamber and turbine?
Materials are one thing. You also need tools and machinery to build one: the turbine and compressor in a mini jet engine rotate at ~100,000 rpm so they're subject to significant stress. They have to be balanced perfectly.
If you want to build one yourself, look into using a turbocharger from a car engine as the core. That gives a decent base for the high-temperature parts of the engine. Others have built DIY jet engines from turbochargers.
GE took on this very task and was able to successfully 3D print a working a working get engine scaled for a model sized aircraft. Although they only claim to use "high strength alloys" and are presumably not going to let the world know what its actually made of. There are lots of easily available small jets for RC planes you may want to look into. There is a nice article here about an engine that utilizes a steel combustion chamber and ceramic blades. You could make one successfully out a variety of materials depending on what you are trying to do.