I like your question, I also find it quite interesting, I think it's different not only between countries, but also within countries, between academic institutions.
I can tell you how it was for my own studies, in one of the universities in Israel.
I actually studied 3 courses in Probability ( not including Stochastic Processes ), during my BA and MA in Statistics. The first course "Introduction to Probability" did not include Measure Theory in any way, it was a basic course in Probability, in high level for Statisticians. The second course was "Distribution Theory", which was a successor course for the first one. Both of these were in my bachelor's degree. At the beginning of my master's degree I had a course called "Introduction to advanced probability", which was a Probability course from the aspect of Measure Theory, with sigma algebras and so on. Hope I helped you