Using a analog multiplexer to use 9 serial input sensors?
No. Not unless you can change your communication protocol to be a "call-and-response" protocol.

Basically, the problem is this: when do you switch to which serial device? Unless all the devices have their timing synchronised there is no way of knowing - so how do you control your analogue multiplexer in that situation...?  You can't.

You would need to have some method of the master switching the multiplexer to a specific peripheral, then send a message to that peripheral saying "Send me your data now". 

You would be better off investigating other boards with more UART ports, such as the Mega2560, and having multiple boards.

The Mega2560 has 4 UART ports. You can have 4 sensors per board (5, if you want to use SoftwareSerial or, better, AltSoftSerial), so you could get away with two boards.

One board acts as normal, communicating on the UART ports to 4 sensors, and also acts as an I2C or SPI master (the choice is yours). The second board communicates to 4 more sensors on UARTs and one further on using AltSoftSerial etc. It also acts as an I2C or SPI slave device.

The "slave" board just runs autonomously gathering data from the sensors it is directly connected to and storing those values.  The "master" then periodically asks the "slave" board for the latest data.