I thought I would come back and tell any future readers of this thread about my experiences. Less than three weeks after I posted this question I got my first invitation for an interview followed by 2 more in quick succession. In total I have been interviewed by 3 companies (a 4th one invited me after I accepted an offer). After a long interview process I now got a part-time job. My advice to anyone who is looking for a part-time job (or any kind of job in the software area) is:
1. Scour job search/carreer websites. Don't limit yourself to jobs.ch and the like but go to others like teilzeitkarriere.ch
or USZ Alumni jobs.
2. German knowledge definitely helps. All 3 companies I have had interviews with were Swiss companies (no multinationals). My CV was in English but I wrote cover letters in German and did the interviews in German. Well, at least partly. They were OK with me answering some questions in English when it became too difficult to express myself in German. I always stated that my German wasn't that fluent at the beginning but people are usually fine if you are up to B2-C1 level and if you say you commit to making the effort to improve your language skills.
3. Do not wait for a job perfectly matching your background to come up. None of the positions I was interviewed for were a perfect match for me. So apply if you meet some of the criteria. If they see you are good in what you have so far done and you have potential, they give you a chance.
4. Be patient. They don't invite you for a first interview until about 2 weeks after you application and they usually give you a date for 2 weeks after.
5. Prepare for the interview. I even looked at some German Software terms. That helped me understand what they were asking me even though I couldn't answer some questions in German.
6. Recruitment agencies are no good. At least they were useless in my case. They were really interested in my CV until I told them I was looking for part-time jobs. All 4 interview invitations came from companies I had directly applied to.
7. Be honest. Tell them what you know, admit what you don't. They really value attitude more than acquired knowledge.
8. Expect a question about your salary expectations even in the first interview.
9. Last but not least, all three of my interview experiences showed me that Swiss people are really nice and can be very warm and welcoming.