When do most students publish in a European PhD?
Probably, you have an incorrect picture of European PhDs.

There are significant differences in the paths that lead to a PhD in the US and in Europe. Several of these differences have been already covered by other questions, but let me highlight those I think are the most important for what concerns your question.

First, however, the usual disclaimer: Europe is not a single entity, and there are large differences among European countries. What I'm going to say below applies to many countries around here, but not necessarily to all.


Most European PhD programmes require an MSc degree for the application. Therefore, the common European PhD student has already done the coursework (or  most of it) that a PhD student in the US has to do during their first years.
Students in the US usually fund themselves through teaching assistantship, which keep them away from research for many hours a week. Most European students, instead, are fully funded by their institution and can work on research full time. Some institutions allow them to work as teaching assistants too, but with a lower workload.
If I got it correctly, US students are allowed during their first years to "wander" a bit to decide what topic they like most. European students are usually not allowed to do so: they start working on a specific topic from week one of their PhD. 
European PhDs are shorter than US ones, but not that shorter: though in many European countries the canonical duration of a PhD is 3 years, the real duration is frequently longer, 3.5-4 years.  


Given the above differences, you should not be surprised to discover that many European PhD students start publishing from their first or second year.

For your last point:


  Or do some people not even make that and wait till post-doc or
  something?


In some cases this would not be possible because some PhD programmes require students to publish at least one journal paper before the end of their PhD.