What will happen to Conservative, Reform and unaffiliated Jews in the Messianic era?
According to Maimonides, who is the only major Halachic commentator to rule on these matters, the Moshiach (the Messiah) will teach all people and encourage them to serve G-d. They will be inspired by his wisdom and will all unite in the service of G-d in harmony. This includes religious Jews, non-religious Jews, and non-Jews.
Furthermore, the Zohar says that Moshiach will bring even righteous to do teshuvah (repent), because one of the primary qualities of Moshiach is humility and he will bring us all to realize that all of us are completely inadequate in our knowledge and service of G-d. (see Zohar III 153b)
וְאִם יַעֲמֹד מֶלֶךְ מִבֵּית דָּוִד הוֹגֶה בַּתּוֹרָה וְעוֹסֵק בְּמִצְוֹת כְּדָוִד אָבִיו. כְּפִי תּוֹרָה שֶׁבִּכְתָב וְשֶׁבְּעַל פֶּה. וְיָכֹף כָּל יִשְׂרָאֵל לֵילֵךְ בָּהּ וּלְחַזֵּק בִּדְקָהּ. וְיִלָּחֵם מִלְחֲמוֹת ה'. הֲרֵי זֶה בְּחֶזְקַת שֶׁהוּא מָשִׁיחַ. (אִם עָשָׂה וְהִצְלִיחַ וּבָנָה מִקְדָּשׁ בִּמְקוֹמוֹ וְקִבֵּץ נִדְחֵי יִשְׂרָאֵל הֲרֵי זֶה מָשִׁיחַ בְּוַדַּאי. וִיתַקֵּן אֶת הָעוֹלָם כֻּלּוֹ לַעֲבֹד אֶת ה' בְּיַחַד שֶׁנֶּאֱמַר כִּי אָז אֶהְפֹּךְ אֶל עַמִּים שָׂפָה בְרוּרָה לִקְרֹא כֻלָּם בְּשֵׁם ה' וּלְעָבְדוֹ שְׁכֶם אֶחָד):
Now, if a king should arise from the House of David who is versed in Torah and engages in Commandments, as did David his forefather, in accordance with both the Written and the Oral Torahs, and he enjoins all of Israel to follow in its ways and encourages them to repair its breaches, and he fights the Wars of G-d, then he may be presumed to be the Messiah. If he succeeds in his efforts and defeats the enemies around and builds the Sanctuary in its proper place and gathers the dispersed of Israel, he is definitely the Messiah.
וכשיעמוד המלך המשיח באמת, ויצליח וירום ויינשא--מיד הם כולן חוזרין ויודעים ששקר נחלו אבותיהם, ושנביאיהם ואבותיהם הטעום.
But when the true King Messiah will rise and succeed, and he will be lifted up and raised aloft, they [the nations of the world] all will immediately return > and will know that their fathers left them an erroneous legacy, and their fathers and prophets led them astray.
וּבְאוֹתוֹ הַזְּמַן לֹא יִהְיֶה שָׁם לֹא רָעָב וְלֹא מִלְחָמָה. וְלֹא קִנְאָה וְתַחֲרוּת. שֶׁהַטּוֹבָה תִּהְיֶה מֻשְׁפַּעַת הַרְבֵּה. וְכָל הַמַּעֲדַנִּים מְצוּיִין כֶּעָפָר. וְלֹא יִהְיֶה עֵסֶק כָּל הָעוֹלָם אֶלָּא לָדַעַת אֶת ה' בִּלְבַד. וּלְפִיכָךְ יִהְיוּ יִשְׂרָאֵל חֲכָמִים גְּדוֹלִים וְיוֹדְעִים דְּבָרִים הַסְּתוּמִים וְיַשִּׂיגוּ דַּעַת בּוֹרְאָם כְּפִי כֹּחַ הָאָדָם. שֶׁנֶּאֱמַר (ישעיה יא, ט) "כִּי מָלְאָה הָאָרֶץ דֵּעָה אֶת ה' כַּמַּיִם לַיָּם מְכַסִּים":
At that time there will be no famines and no wars, no envy and no competition. For the Good will be very pervasive148. All the delicacies will be as readily available as is dust. The world will only be engaged in knowing G-d. Then, there will be very wise people who will understand the deep, sealed matters. They will then achieve knowledge of the Creator to as high a degree as humanly possible, as it says, “For the Earth shall be filled of knowledge of the Lord, as the waters cover the sea” (Isaiah 11:9). Blessed be Hashem who helped me.
וְהָיָ֣ה ׀ בַּיּ֣וֹם הַה֗וּא יִתָּקַע֮ בְּשׁוֹפָ֣ר גָּדוֹל֒ וּבָ֗אוּ הָאֹֽבְדִים֙ בְּאֶ֣רֶץ אַשּׁ֔וּר וְהַנִּדָּחִ֖ים בְּאֶ֣רֶץ מִצְרָ֑יִם וְהִשְׁתַּחֲו֧וּ לַיהוָ֛ה בְּהַ֥ר הַקֹּ֖דֶשׁ בִּירוּשָׁלִָֽם׃
And in that day, a great ram’s horn shall be sounded; and the strayed who are in the land of Assyria and the expelled who are in the land of Egypt shall come and worship the LORD on the holy mount, in Jerusalem.
כִּֽי־אָ֛ז אֶהְפֹּ֥ךְ אֶל־עַמִּ֖ים שָׂפָ֣ה בְרוּרָ֑ה לִקְרֹ֤א כֻלָּם֙ בְּשֵׁ֣ם יְהוָ֔ה לְעָבְד֖וֹ שְׁכֶ֥ם אֶחָֽד׃
For then I will make the peoples pure of speech, So that they all invoke the LORD by name And serve Him with one accord.
Welcome to Mi Yodeya.
Your question is a good one and something that everyone, including every Jew should be thinking about. In fact, it is meritorious to have the subject of Moshiach and his imminent arrival in you thoughts each and every day like is said in regard to Moshiach, ״and, though he tarry, I watch for him all day that he will come...״.
This is not just in a general sense, but also in regard to the details.
We follow the teaching which we received from Yehoshuah ben Nun, the student and successor to Moshe Rabbeinu about this. This teaching is said by Jews at the close of each daily prayer service with the Aleinu prayer. It says:
that You will sweep idolatry away so that false gods will be utterly destroyed, and that You will perfect the world by Your sovereignty so that all humanity will invoke Your name, and all the earth's wicked will return to You, repentant. Then all who live will know that to You every knee must bend, every tongue pledge loyalty.
It does not say that destruction comes upon humanity, G-d forbid, but rather that everyone will be perfected through Moshiach's coming.
Nobody knows exactly until it happens, see Rambam end of Melachim (cited above). Don't believe anyone who says he knows, it is all just speculations.
Keep in mind that the "time of Moshiach" is a very long era spanning probably a hundred of years from his coming to the building of the Temple and slow transformation into the World to Come, which nullifies the free choice and everyone becomes a robot (sort of).
You probably expect the world to divide into righteous and wicked, and at "the end of the world" - the Messianic times the pious getting all the benefits and rewards and the wicked get punished. This idea has some roots in Judaism, but I would like to focus on the definition of the pious and the wicked.
We have two systems for measuring people's "religious" progress: an absolute and a relative:
The absolute system takes the contemporary Jewish Law (Shu"A and the latest Poskim) as the baseline and the person's behavior is assessed against it - the more he observes it the more reward he gets and the more he overrides it the more punishment he has.
The relative system assesses the person according to his personal destiny and the personal goal of his existence. It might be learning Torah and it might be one particular Mitzvah (such as honoring parents. See my other anser). According to this approach, a person might be a pious Jew observing the whole Shu"A but missing his goal and the opposite.
Therefore according to the second approach, we can't judge the people based on their appearance, they might be looking secular and be pious in G-d's eyes or the opposite.
All the previous answers focused on the first approach, while it seems that the Mashiach's mission (to bring the world to its completion) is to teach everyone his personal challenges and purposes resulting in what the Gemmorah allegorically hints (Yevomos 62a): "אין בן דוד בא עד שיכלו כל הנשמות שבגוף"