I am second in turn order, I select Urbanization. Do I place the city at the start of my build track phase or before the first person who builds track?
You place the new city as your first action during your Build Track phase.
On page 2 of the rules
Implemented during the Build Track phase. This action allows this player to place one of the New City tiles on a Town before they build their track.
This is from the Warfrog edition (2002). It seems the rules haven't changed much at all despite the plethora of editions.