What interests me is two fold:-
a) the 'wand', how the inductors are wound, their size and the interaction with the coupling metal.
b) how would one discriminate between different metals?
My two cents:
a) Not critical. You could use a ferrite rod from a loopstick antenna from an AM BCB receiver with similar results. It looks to me like a Hartley oscillator circuit, you use enough windings in the tank circuit to produce the frequency you desire, and the other winding is a feedback circuit that creates the oscillation. The number of turns in the "tickler" winding depends on how much positive feedback is required. Dave was not specific on the operating frequency but you can determine that from the measured inductance and the parallel capacitance of the tank circuit. This is not a field balancing coil so you should be able to wind the tickler winding directly over the tank.
This circuit operates similarly to a dip meter or even a regenerative receiver - the presence of the metal detunes the oscillator, reduces the Q of the circuit and absorbs a portion of the generated RF. The detector output decreases and once it goes below the opamp comparator's 2.5v threshold it energizes the piezo buzzer.
b) This circuit only detects the amplitude of the oscillator output. To detect the metal type, the frequency of the oscillator must be monitored with a frequency discriminator, hence the need for a second oscillator to act as a beat frequency oscillator. The different types of metals have different effects on the tuned circuit. Ferrous metals cause the frequency to decrease (increased permeability of the inductor) while nonferrous metals like copper, silver and gold have the effect of decreasing permeability which reduces the inductance of the LC circuit and causes the frequency to increase.