Do you re-determine player order in this exceptional case?
If you are the only one to build no cities in round 1, you will go last in round 2.

The determine turn order step in round 2 is not straightforward at all; in most games I see, players build different numbers of cities in round 1, ranging from 1 to 3.  That your play group has been skipping the turn order step in round 2 means you have probably been playing wrong in one way or another.

My best guess of the mistake you are making is it sounds like you are only allowing players to build a single city in round 1.  Players can build multiple cities in round 1 (there is nothing in the rules that prevents you from buying more cities on the same turn you buy your starting city).  Most players usually do buy multiple cities on round 1, as many of the early power plants can power more than one city.  The adjusting of turn order in round 2 is an important mechanic to balance against this: players who buy a power plant that can only power one city in round 1 can then build only 1 city to get the a better turn slot on round 2 (later in turn order is better) to make up for their lost income.