No problem.

OK, then you wouldn't have access to GPUs anyway. (Buying them is expressly prohibited by UA, and if you get caught, you can wind up sitting in the back with your upgrade cancelled and your MileagePlus account closed). Furthermore, even if you had a benefactor who were willing to give you some GPUs as a gift (which is allowed), the fact that you don't have status and you're flying on the cheapest fare class that UA offers on trans-Atlantic flights means that your chances of having the upgrade clear are minuscule -- you'd basically be last in line.

If there were a cheap way to upgrade, nobody would sit in coach. Your best bet is to keep your fingers crossed and hope that the upgrade price drops at some point prior to departure. If you have a smartphone, install the United app, if you haven't already, so you can monitor the upgrade price right up until boarding. They will absolutely try to sell the seats. Even though your wife is traveling with miles, there's a good chance that she'll get a cash upgrade offer at check-in too.