So, I would like to share how I handled the situation, to help people in future.
First thing I should have done was to go to Indian Embassy in Vienna on the same day (Emergency services are open on weekends as well). But I didn't know that, so the bus drivers let us board the buses from Vienna-Bratislava and Bratislava-Budapest during the weekend, based on our police report, and I also told them that we have an appointment at the Budapest Embassy on Monday for issuing a temporary passport.
I arrived in Budapest and went to the Indian Embassy there on Monday, they asked us to fill a long form, and some passport sized pictures. They issued us the passport on Tuesday and we were ready to go to Italy on Wednesday. BUT. The old guy at the embassy was not very helpful, he kept telling us that we should just go back to India and the temporary passport is not for travelling within the Schengen.
I decided to take the risk because we had a long trip ahead of us, and also Schengen is a single region, we were technically not leaving our region mentioned on the Visa. I called up the Indian embassy in Italy to confirm, and they said that it's usually not a problem but they're not sure. It went smooth on the Italian border because no border control is there. Our next destination was France, which had a very strict border control, so I was preparing myself to send my Mom back to India because French officers won't approve of the Emergency Certificate(temporary passport). But the border control officer was so great, he didn't even utter a single word, he just nodded when we showed him the EC.
The rest of the trip was smooth, and my Mom went to India 20 days later as planned from Germany, and now she has her new passport in India.
- Always leave your passports at the hotel and keep a copy of them and the Visa with you all the time.
- Contact the police, get a report, make a lot of copies, and then contact the Emergency service of the Embassy of that country.
- In my opinion, you can travel with the EC within Schengen. No embassy official will tell you that, because it's not clear to them, and honestly it depends on the border control officer.
Most important: Don't panic!