lcd 5110
If you are sure that both connections (Rx and Tx) are correct then the only one thing that can be, and probably is, suspicious is the supply voltage, 3.3V that is ..
To connect LCD you have to lower the standard supply from 5V to 3.3V ..
In normal circumstances even 0-5V for RS-232 standard wouldn't be enough, but nowaday's computers can "understand 0-5V levels (instead of defined by the RS-232 standard: -25-3 and +3+25) ..
So, here you have the microcontroller supplied of 3.3V and you still expect the RS-232 port to work correctly?!?!?!
It is quiet possible the PIC (PICAXE in this case) can receive transmission from the PC but what it tries to send back its feedback (acknowledgment) it is just below acceptable level and therefore your PC tells you: NOTHIG CONNECTED ..
To program PICAXE disconnect LCD, connect 5V supply to PICAXE and try again ..
Regards,
IanP