What is going to happen to all the unobservant jews (tinok shinoshbas) Right when mashiah arrives? Will they all start observing the torah? Will they then still get to go to olam haba?

The sages use the term Y'mot HaMashiach [=the Days of Mashiach] referring to a “period” and process, and not one magical super-natural day, as you inferred: "right when Mashiach comes". Modern times have clearly shown, that God runs Jewish history and implements the redemption: gathering the exiles, reviving Hebrew, the Israeli desert and army, through logical, natural ways harmonized with human "free will", e.g. the pioneers chose to return to Israel for national reasons or anti-Semitism, and not necessarily "religious" reasons. Similarly, one sees statistically that the State of Israel is already gradually, but consistently and democratically (!), becoming more religious, as seen in the religious members of Knesset, our first religious Prime-Minister, and even more so, in registration for religious education, youth movements (the largest are the religious) and programs for ba'alei tshuva. The traditional have a much higher birthrate, as well as Aliya-rate from western countries (including France), as opposed to those "yordim" unfortunately leaving Israel, who are mainly irreligious. We pray and believe that the "snow-ball" effect, and unforeseen joyful Godly events, will hasten this phenomena of spiritual rejuvenation even more, in our lifetime. Nevertheless, and as indicated above, these changes must be voluntary (democratic!) and not through religious coercion; otherwise, they are not real, happy nor lasting. In addition, it's important to remember that the ge'ula is for all of Israel, and all mankind, and not just for those who are already religious. In fact, the democratic process is ideal for such changes to be made according to the pace chosen by the electorate. The legendary Chief Rabbi A.I. Kook explained in this context, that true improvement is the result of educating each generation better and better, and certainly not through religious coercion which causes animosity, rather than identification. The democratic and educational process is by far the most natural, real and long-lasting, and that’s exactly what we (and G-d) all want. Although G-d can do anything, already in the Book of Esther, about 2,500 years ago, He clearly prefers not doing super-natural miracles, but rather working through nature (like the Six-Day War in ‘67), thus enabling us to believe in Him through our free will and not through miraculous coercion. The essence is to arrive at the national and spiritual level of redemption, which is a long process, and not magic, overnight or supernatural. An essential part of this process is when all will see the aforementioned fulfillment of the prophecies of national and spiritual revival, which will slowly convince even the skeptical. At the end of the process, Mashiach be David will arrive (Megilla 17b). Regarding the world-to-come, all Israel has a share there, but some have a larger share than others, and the aforementioned teshuva (return to Torah) surely fixes the previous mistakes of observant and non-observant alike.