Okay so I've figured it out, when communicating with the onboard SARA-U201 GSM module I've notice that the CNMI parameter was set to 1,0,0,0
AT+CNMI?
AT+CNMI=1,0,0,0
I did some research on general SMS settings and I found out that the CNMI needs to be configured to 1,2,0,0
AT+CNMI=1,2,0,0
I've setup up the Arduino to program this at the end of the setup function.
Then, when a SMS is received by the module, it is automatically transferred to the Arduino, usually within 4 seconds. I then wrote a piece of code that reads the GSMserial and parses the message to extract the content of the SMS and getting rid of the rest.
I assume that the sms.avaialble() was polling the GSM module at a regular interval, which is why it was way slower than this new configuration...