Unfortunately, bipolar junction transistors (BJT) won't work as you intended here.
I believe you are assuming that a BJT, when switched on, is like a very low resistance between collector and emitter, but this is not the case. BJTs are ill suited to the task of forming switchable high/low impedance paths from input to output, because their base-emitter junction is effectively a diode, which ties the base and emitter potentials to within 0.7V of each other.
Therefore, in the configuration you have drawn, the emitter always "follows" the base signal, not the collector. in fact, it's called an emitter follower" for that reason, which you can read about in the Wikipedia article.
What you require is either some kind of MOSFET design, or better, an analogue switch, which does the job for you.
Even MOSFETs will require carefully crafted potentials. Even though they are more like the switchable impedance drain-to-source path you require, MOSFETS have a parastic body diode, which makes that simplified model of their behaviour only true for certain gate potentials. Also, for a MOSFET to be "on", it relies on the gate potential being very different from the potential of the signal being switched, which is not trivial to achieve.
For these reasons analogue switch ICs typically do not use BJTs as signal path switches. They make use of two or more MOSFETs to get the job done, and they have all the additional circuitry necessary to ensure that gate potentials are appropriate for any given signal potential.
Analogue switches, when "on" provide signals paths of just a few Ohms, or milliohms in some cases, and that path is fairly independent of the switching circuitry. However, due to the gate potential constraints I mentioned, they all require that the analogue signal potential stay between the power supply potentials. Analogue switch ICs usually have three power supply pins, ground, positive and negative. The signal being switched cannot ever exceed either the positive or negative supply voltages.
Another cause for concern is amplitude of the analogue signals themselves. If they require amplification, then that should be done as early as possible. Every stage they traverse in some system (even an analogue switch) is adding noise to the signal. By amplifying at a late stage, each prior stage is adding small amounts of noise to a small signal. By amplifying first, each subsequent stage is adding a small amount of noise to a large signal. Obviously, in the latter case signal-to-noise ratio is much better.
Lastly, don't rule out relays. They provide complete isolation of analogue signal from the switching system, and don't suffer the amplitude restrictions imposed by analogue switches.
So, I have two recommendations:
Amplify first, switch later.
Use an analogue switch IC or relay.
Some analogue switches you might consider are: