If the main shock wave is formed at the nosecone of a jet aircraft at supersonic speeds, how is the airflow still supersonic at the inlet of the jet intake? Shouldn't the inlets experience subsonic flow since air downstream of shock will be subsonic?
When it is said that "flow is subsonic behind the shockwave", it is only the component of velocity perpendicular to the shockwave which is actually subsonic.
In a normal shockwave, this makes no difference - the net velocity is the same as its perpendicular component. If the perpendicular component is subsonic (which it always is), the net velocity will inherently be subsonic.
But in an oblique shockwave, the net airflow velocity behind the shock can be supersonic, even though its perpendicular component is subsonic.
In the picture that you shared, you can see that the shockwave is almost entirely oblique (the normal shock at the nosecone is so small that it cannot even be seen in the picture). Anyways, the net velocity of flow behind this shockwave is still supersonic, which means that it will again produce a shockwave when it interacts with the inlets and other parts of the aircraft.
In the Schlieren photograph you present, you can see that an oblique (slanted) shock is formed wherever there is a change in the cross-section of the aircraft profile, illustrating Aditya Sharma's point that the flow velocity behind an oblique shock is supersonic. Only after traversing a series of oblique shocks does the flow behind the last such shock become subsonic.
This principle of exploiting a series of oblique shocks to progressively decelerate airflow to below mach 1 was used in the design of the J58 inlet structure on the SR-71. In principle, such a series of progressive shocks (which were positioned between the shock cone tip and the first-compressor stage at the face of the J58) can furnish near-perfect ram pressure recovery (lossless compression), thereby performing almost all the compression work required to drive the afterburner of the J-58 during mach 3 cruise.