What should I do to get into a PhD in Mathematics in Europe without a thesis in my BSc and MSc?
First - do you think I'd have any chance without doing some sort of independent research before applying to a PhD? 


Possibly. However, it would likely involve you finding some "angle" on the basis of which you can strike up a relationship with a prospective supervisor, who could promote/sponsor your Ph.D. candidacy.

But there's a deeper issue here:

Why do you want to enter a Ph.D. program?

Hopefully, you have some subject you are interested in studying. If that is not the case, and your motivation is to simply get the title and the piece of paper - then you should seriously reconsider. Working on a PhD without focused motivation is not just exceedingly difficult, but also exasperating. You may well find yourself in the middle of it, two or three years from now, grabbing your head by your hands and asking "What was I thinking? Why in blazes am I even doing this?" - and not having a good answer.

So, do consider some independent research, or even going into industry doing something you are interested in, but with a mind to sketch out relevant mathematical challenges to study more deeply in academia. These are my suggestions for filling the gap.


  my top choices are Germany, Switzerland, UK, and France


Once you find something relevant to research, locate relevant research groups in that field (in those countries or elsewhere), and talk to them.