T1 forms a colpitts oscillator: -
Image taken from this article.
T2 is a buffer amplifier in common-collector mode - it feeds an amplitude detector formed by D1 and D2.
There will be a varying DC amplitude across C7 as the search coil has energy stolen from its magnetic field by metal objects it encounters. This voltage when it rises will activate T3 and reset the 555 timer.
The 555 timer looks like it's operating as an astable multivibrator.
So, it seems that when you encounter a piece of metal, either the oscillator stops or starts depending on the type of metal - ferrous will do the opposite to non-ferrous.
BTW it's a really ineffective design unless you are looking to detect large objects.
If the coil has a high impedance then it's probably gone open circuit. Try measuring the DC resistance.