Though the oblique shocks reduce the downstream mach number, the flow will still be supersonic usually; for the flow to be subsonic, the shock has to be either normal or detached (bow shock).
For every body angle (i.e. corner angle)-mach number combination, there is a maximum corner angle $\theta_{max}$, beyond which the shock will be detached from the body.
![Oblique Shock](https://i.sstatic.net/uJwJ5.png)
Oblique shock angle, β, as a function of the corner angle, θ; By -- Myth (Talk) 05:29, 27 October 2007 (UTC) - Own work (Original text: self-made), Public Domain, Link
For cases where the corner angle is less than the maximum, there are two solutions possible- strong and weak. The weak solution leads to downstream supersonic flow (and smaller shock wave angle β), while the strong one leads to subsonic flow downstream (and larger shock wave angle β).
Which solution is 'preferred' depends on the ratio of upstream and downstream pressure; in case of external flows which concern us, downstream pressure is usually close to upstream pressure (both near $P_{atm}$) and as a result, the weak solution (and downstream supersonic flow) is 'selected'.
From NACA Report 1135: Equations, tables and Charts for Compressible Flow:
... two solutions exist for each cone and Mach number, but it is believed that only the weaker shock wave can occur on an isolated convex body.
So, the wings have to be designed for supersonic regime in case the nose produces oblique shock waves (detached shock waves are no good- they increase drag tremendously and that is the main reason they are used in re-entry vehicles). Even if the flow is not supersonic, you're still in transonic regime, where you need design the wings keeping the critical Mach number in mind.
Though the 3D flow over a cone is similar to the flow over the wedge (in that it has strong-weak cases and separation above a certain limit), the maximum angle $\theta_{max}$ is higher in case of 3D flow. Again, the attached shock is the 'weak' one.
Because of the 3D relieving effect (which causes a weaker shock), the pressure on the cone surface is less than the wedge surface pressure and the cone surface Mach number is greater than that of the wedge surface. As a result, the designer has to contend with supersonic flow still.