Would this minimal NPN-based metal-detector circuit work?
No - or at least not like you expect. I was writing a long explanation of why not and the physics, but then I accidentally pressed a button and lost it all...

So instead I'll point you to two things you need to understand:

Eddy currents: http://en.wikipedia.org/wiki/Eddy_current

How Metal Detectors Work: http://electronics.howstuffworks.com/gadgets/other-gadgets/metal-detector2.htm

Pay attention to the fact that it produces an AC field which makes eddy currents come far more naturally.

Your circuit will produce a DC loop.

Technically, it will work... but only as you're moving it [quickly] and only with tiny, tiny, tiny current fluctuations making tiny, tiny, tiny sounds.

For reference, these forums generally assume you know at least some of the physics/mathematical/electrical background of the subject matter, but it sounds like you have only an idea that "transistors and coils do stuff." I'd strongly recommend finding old textbooks on these subjects and reading more about it, starting with an overview of Physics Electricity and Magnetism and going to the abstractions of AC/DC fields and inductance, then transistors.