The important idea here is that God is eternal, and even should (?) our world cease to exist, He will remain, as He was before He created it.
The prophets don’t refer to any end of the world (contrarily, there’s room to understand the covenant with Noach that it won’t happen). There are varying traditions, and each one would explain the line you’re referring to, accordingly. The Talmud (Sanhedrin 97b) cites several possibilities, one of which is that the world as we know it, will exist until the year 6,000 (in another 225 years!), and the Rashba (Resp. I, 9) accepts this as an ancient Jewish tradition. On the other hand, the Rambam (Maimonides, Guide for the Perplexed, 2, 29) differs and opines that our world is eternal. He explains that as many midrashim, the above opinion isn’t meant to be taken literally and isn’t meant as historically accurate, but either symbolic or maybe simply an individual opinion, and not “mandatory” to accept. In short, there are several different traditions, and our rabbis tell us there (Sanhedrin 97b), that one isn’t allowed (!) to base his plans or decisions on any of them, for maybe another is correct!
The simple meaning of the aforementioned midrash is that the world is like the work-week, where each day of creation corresponds to a 1,000 years (see Tehilim 90, 4), and then we will enjoy the redemption which is likened to a “world of Shabbat”. Note, that really every week, we are supposed to prepare and “accept Shabbat early” and not wait for sundown (=the year 6,000, based on Vilna Gaon). Similarly, we could and should “move up” the final redemption, as well (historically, today already corresponds to “Friday afternoon”!) and enjoy and be thankful for that which has already begun .
The messianic world will not change anything in nature, but just that independence will return to Israel (Sanhedrin 91b, Rambam Tshuva 9, 2).
Regarding the nuclear threat which you may be alluding to, no need to worry! Our rabbis teach us in many places the ancient tradition that once we will return the third time to the land of Israel (the first was from Egypt, the second from Babylon with Ezra), there will not be another exile (Tanchuma Shoftim 9, Yalkut Shimoni, Hoshea 518). In the time of Ezra, the 42,360 who came back are considered a “return”, so how much more so, today, when so many more Jews have already come home (6,200,000), that we are undoubtedly witnessing the third and final return, and that we’re here to stay, as promised. Our ancient rabbis actually dealt in the Talmud explicitly with the modern threats, and tell us not to fear (Sanhedrin 106a). To the contrary, the enemies of Israel should worry about themselves:
“Woe to the nation who tries to prevent the Father from redeeming His son (=Israel)! Woe to the fool who interferes with the Lion when He wishes to mate with His beloved lioness (=Israel, after such a long 2,000 year separation)!”
We aren't allowed to rely on miracles, so our enemies will probably stand to “get blasted” from both God and the IDF, the Lion and the lioness, working once again, together as in the days of Moshe, Joshua and David!
“And the main point, is not to fear at all!” (R. Nachman)