The problem with the Serial Port has now been fixed, or at least, I have a workaround. Bill Rowe suggested I activate the /BUSRQ line to force the Z80's Address and Data lines into High Impedance. After soldering the switch line on to the /INT pin, directly opposite the /BUSRQ pin on the Z80 and being frustrated for a number of hours before finding the mistake! I re-wired it with a 10K pullup resistor and it works fine. Pulling down the /BUSRQ line does indeed allow the serial port on the AVR to receive programming from the Arduino IDE. The /BUSACK line also goes low to indicate this.

I'll have to think of a solution for the next version of the board. Could I drive some sort of circuit off of the DTR line on the serial port? Or will I have to have a switch on the board to select between Program or Run?