Short answer: once they contact you, expect everything done in 1-5 days.
I applied online back in March, but didn't hear anything until early May. At first I was sad because I made some writing mistakes on resume and cover letter and I thought that was it for my internship application. The recruiter reached out to me later and said they were just busy reviewing. I also apologized to them.
The recruiter scheduled my first interview that week immediately. It happened on Friday on Skype. I first met with the manager of the team. He basically asked me to describe myself, my work, and why I was interested in Mozilla and the team. Then we tried to schedule two more technical interviews immedaitely on the same day. However, only one was able to carry out. Turns out the engineer didn't prepare any coding problem as he thought coding problems don't really tell much about the quality of the intern. We chatted a bit about security and I got to ask him his experience at Mozilla and how he ended up in Mozilla. It was a fun chat.
On Monday I was given the last interview. I thought it would be a real technical interview but again it was just another "get-to-know-you" interview. We went out strengths and areas I would be interested in should I get the offer.
I did a lot of research about the team prior to job application and after sending in the job application. I basically read their Wiki and github page. I read projects from other teams to know more about what people are doing at Mozilla. That has proven to be really really helpful so I recommend future interns and future full-timer do that. Research!
So the evening of the same day I received confirmation and I accepted the offer officially today. I probably will be known in a few weeks if my team checks this website lol. I consider myself a lucky person, but at the same time I had to spend a lot of time learning about the position and Mozilla, as well as building the basic skillset I need for this internship.