Turbojet can't be used on subsonic cruise missile. Turbojet and turbofan are extremely different. Turbojet is you traditional jet motor which uses solid fuel and solely relies on the thrust from the fuel ejection to fly. Due to this, high speed can be achieved by keeping such kind of fuel that is expelled quickly. Ramjet is another form of Turbojet which uses solid fuel but has no solid oxidizer mixed with the fuel. Instead, oxygen is sucked from the air to burn the fuel.
Turbofan is an engine that burns liquid fuel like kerosene, ether etc to rotate the fan and not eject fuel with high momentum. Something like a helicopter fan but smaller. The fan sucks the air and then pushes it back with greater force to propel the vehicle. Here, the fuel is not ejected directly but the fan merely pushes the air back faster - called bypass. The cruise missiles have high bypass while jet engine has low bypass. Cruise missiles are more fuel efficient than jet aircraft. The after burner in jet engine works something like a turbojet by wasting huge fuel
The result of this difference is that turbofan can't fly much faster and is subsonic. Turbojet, however works well when supersonic.
Nirbhay couldn't have been turbojet at all. It is mandatory for it to be a turbofan. Turbofan for cruise missiles have no link with Kaveri engine. Though both are turbofan, Nirbhay uses 4kN thrust while Kaveri is 90kN. The difference in thrust is so huge that Nirbhay turbofan is far easier than Kaveri engine. The problem of single crystal blades, melting points, super alloy etc doesn't arise. 4kN thrust has minimal temperature or pressure, even for the smaller size of Nirbhay and hence easily accomplished.
There should be no doubt that Nirbhay used Turbofan engines