Can I fit a 32-630 tire on a 20-622 rim? I know that the width difference might present problems later on, but if I understand correctly then it should be possible. The bike isn't meant to be used in high performance circumstances, just the city. 

However, if I understand correctly, the tire would have a larger diameter than the rim. Is that likely to cause me much trouble?

tl;dr If I have a 32-630 tire and a 20-622 rim, can I make the combination work?
No.  The bead seat diameter (inner diameter) is different by 8 mm.  You cannot shrink a steel or Kevlar bead by 8mm, as they are designed not to stretch  or shrink (otherwise a clincher tire will roll off the rim in a corner).   You can try but the tube will burst out from the extra space. 

The tire bead is what interfaces with the rim.