I've just tested the appointment system for Alicante province for an NIE by a British national.
The only location it offers is at Torrevieja, but when I ask for an appointment it says:
NO HAY SUFICIENTES CITAS DISPONIBLES
Lo sentimos, pero en éste momento la Oficina no puede ofrecerle una cita para atender su solicitud.
Out of interest, I got exactly the same when I said I was German.