This issue is caused by the default skybox reflection cubemap not being updated after the skybox material is changed. There is currently no functionality to request a rebake of said texture at runtime directly, which of course isn't ideal. We will treat the issue as a feature request and prioritize it accordingly.

One way to work around this is to not rely on the default reflection cubemap, and instead use reflection probes to provide reflections to all renderers. Another workaround, is to bake a skybox cubemap yourself and assign as the currently active one via RenderSettings.customReflectionTexture (make sure Environment Reflections Source is set to Custom). One way to bake such a cubemap in practice is to use a realtime reflection probe with culling mask 'Nothing' and Refresh Mode 'Via Scripting'. Calling ReflectionProbe.RenderProbe() will rebake the cubemap for this probe, and that cubemap can then be assigned to the field in RenderSettings.

See the example MonoBehaviour below for more details. One may call the UpdateEnvironment coroutine to rebake all environment lighting, for example after having changed the skybox material.

public class UpdateSky : MonoBehaviour {
ReflectionProbe baker;

void Start() {
baker = gameObject.AddComponent<ReflectionProbe>();
baker.cullingMask = 0;
baker.refreshMode = ReflectionProbeRefreshMode.ViaScripting;
baker.mode = ReflectionProbeMode.Realtime;
baker.timeSlicingMode = ReflectionProbeTimeSlicingMode.NoTimeSlicing;

RenderSettings.defaultReflectionMode = DefaultReflectionMode.Custom;
StartCoroutine(UpdateEnvironment());
}

IEnumerator UpdateEnvironment() {
DynamicGI.UpdateEnvironment();
baker.RenderProbe();
yield return new WaitForEndOfFrame();
RenderSettings.customReflectionTexture = baker.texture;
}
}