Short of posting the sketch to this forum somewhere, is there anything that springs to mind that I should be looking for?
The most common source of Uno Vs mega problem is for attached things using the I2C pins or the SPI pins as their location is on different pins for those boards. But if your not using either of those interface methods then no, nothing pops out at me, so post your code.
Correct, I am not using I2C nor SPI. The TXD data from the module is being read on pin D2 as INPUT with the pull-up resistor set on.
My ATMega328 is built on a RBBB board; the ATMega2560 is a Cappucino.
Code in attached files...