In this application, stray capacitance and reverse recovery time are going to be more important than forward drop. I would have thought that the 1n914 would be ideal. Though, 1n4148 should do.
Other point is that the LED driver, having no base resistor, is going to set a hard limit on the rectified voltage equal to its Vbe drop. This will load the oscillator heavily. Not sure if that is intended, but I would have thought not.