Difference between TGC (Time Gain Compensation) and AGC (Automatic Gain Control)

You need to rectify and filter the output, to get a DC signal that varies with the output level. A single-stage amplifier is often used before the rectifier. The DC signal is used to provide feedback to a gain control element at the amplifier input. Something like a FET is often used, with the feedback signal amplified by an op amp. The feedback has to be negative, of course. The system will need some tuning, to avoid problems with the gain rising when there are quiet periods, so that the first syllable of the next word comes out at high volume, before the AGC kicks in.

Your voltage to current convertor is inappropriate. With no speech signal a current flows into Q3 of approximately 12mA (\$\frac{12 volts}{1k\Omega}\$). I'd try making R3 and R4 much larger (try 10k for a start) because the "long-tailed pair" you have needs to operate in the saturation (linear resistance) region to work correctly: -

Operation in the saturation region means your transistors can operate similar to a current controlled resistor BUT your CE voltage needs to be low and you need to look at transistor data sheets to see how low. \$I_B\$, in effect picks the resistor value (which is the slope on the curves above). Varying I\$_C\$ in the linear region varies V\$_CE\$ because it is like a resistor and because \$I_B\$ is varying the resistance you get amplitude modulation or basic 2-quadrant multiplication BUT collector-emitter voltage needs to be low or you enter the right-hand side of the graph which is the part for normal BJT amplifiers.

Second problem is that your speech input will actually modulate your music input because the current into Q3 will rise and fall (centred at 12mA) with the signal shape of the voice. You need to track the amplitude or envelope of the speech signal and I can't make any recommendations on this other than use a precision rectifier circuit built around your first op-amp OA1.

I would also point out that OA2 has no local feedback, relying on OA3 to provide negative feedback. This is a recipe for oscillations and instability. Also, I note that you have local positive feedback around OA3 and in conclusion I would ask, where did you get this circuit? On the face of it, has has smatterings of the right approach but every section is flawed with mistakes and inappropriate design.