Can I use an axial inductor instead of a radial inductor?
The proposed ones will have a magnetic core only a different shape from the originals - there is no way that they can get 4.7mH without one.  The original inductors are wound on what are called Drum cores.  Your proposed one probably just has a cylindrical core.  Since the magnetic field is not as well contained it will probably need more turns of finer wire which will increase its resistance and make it more lossy.

If the resistance of the proposed inductor is significantly different from the original the characteristics of the filter will be different.

Try simulating it in LTspice to compare the effect of any resistance change between the original and your proposed inductor.

I would expect it to perform adequately with the proposed parts.