Have you checked to see if the mute function is activated? Make sure that the microphone is not muted by pressing the mute button on the adapter.
It's also possible that the headset is not completely connected to the adapter, or the adapter may not be fully connected to the controller. In this case, the speakers would not be working as well.
If these suggestions did not solve the problem, check out the [[Xbox One Stereo Headset Troubleshooting|troubleshooting page]].