Full disclosure I have not opened my Nissan EVSE and am not an EE (although I play one on TV)...
Pictures from http://www.mynissanleaf.com/viewtopic.php?f=26&t=2633
I have to agree with lincomatic, the optimal way to upgrade the Nissan EVSE and likely what Phil at evseupgrade.com is doing is to replace the 120V only transformer (2nd picture) with a universal switch mode power supply. Doing so will change the output from AC to DC requiring depoting of the board to bypass the rectification diodes. There may or maybe not be other components which require upgrade due to the voltage increase. As mentioned in other posts the relays do not require upgrade as they are rated at 250V @ 20A (picture 4).
Powering the mainboard on 120 and bypassing 240 to the relays is a bit messy, while it would work I wouldnt do it...Doing it the right way may cost $30 instead of $20 and all the safety features would remain.
I believe the ground check uses the 3rd smaller relay (3rd picture far right) to pass a small current from L1 Hot to ground through a 5k or so resistor. If you listen carefully you can hear a clicking right after connection before the main relays engage. I do not think it is checking to see if Neutral and Ground are tied together only that there is a path to ground. Routing current through the 5k resistor would be enough to cause a ground fault so.... my theory is the EVSE powers the 3rd relay causing current to flow to Ground causing the GFCI to read a fault if so, ground and the GFCI CT coil is good. If ground is bad no current flows and the GFCI does not read a fault and the EVSE goes into Fault state.