No rushing noise out of speaker
The first thing is to disconnect the BFO tuning capacitor at the BOLD rectangle marked "BFO Tune". If you have accidentally grounded the BFO tuning capacitor, removing the "BFO Tune" connection will remove the tuning capacitor from the circuit.
Retune the BFO transformer for a rushing noise out of the speaker. The transformer will tune to 455 kHz without the variable capacitor.
If you get the rushing noise now, look over the BFO variable capacitor to find the short to ground. Look for a splash of solder between the plates or you could have soldered the stator and rotor together.
If this does not solve the problem, make sure the LED is on at the BFO amplifier. If not, adjust the "Gain Adjust" trimmer or check for a bad solder joint or missing part.
One way to tell if the oscillator is working is to replace the 47pf coupling capacitor between the BFO amplifier and the product detector with an LED. Power up the receiver and if the LED lights, then the oscillator is working.
If the LED does not light, check the BFO circuitry for a bad solder joint, 5 Volts to the 100 ohm resistor on the output of the 7805, or a missing part.
If you still don't get noise out of the speaker, then check out the audio amplifiers.