Using UART's TX & RX for Separate Functinality
What you want to do can be done, but there are limitations

1) A uart runs at one speed, so you will have to run RX and TX at the same speed.  

2) Make sure the logic levels are compatible.

3) Make sure that both the PC and the GSM have the same uart setup, ie stop bits partity bits ect.  

Trouble shoot them one at a time. Note the settings and when you get them both working independently then combine them. I would make life easy on yourself and put them on separate uarts.