How does inductance change with presence of different directional magnetic fields?
The moving magnet will induce a voltage in the coil. If the magnet is fixed there will be some effect on the inductance due to eddy currents in the magnet (assuming it is conductive). If you have a non-conductive magnet, then there will be no effect on the inductance of an air-core inductor. 

If there is a core, the inductance is a nonlinear function of the current (and the magnetic field) and an externally applied field can bias the core and (most likely) reduce the inductance.

It's not easy to calculate these changes, you'd need a field simulation program like COMSOL.