When was Ba'bel built and what became of Nimrod after that?

Also see BABEL - Links to Information From the Watchtower Online Library)

According to the Bible, when was the city (and tower) of Ba'bel built?

The date of the attempt at building the city of Babel is not stated in the record, but Genesis 10:25 indicates that the division resulting from the confusion of the languages there occurred sometime during `the days of Peleg'. His name even meant “Division,” for “in his days the earth [that is, “earth’s population”] was divided”; Jehovah “scattered them from there over all the surface of the earth.” (Ge 10:25; 11:9) A text of Skarkalisharri, king of Agade (Accad) in patriarchal times, mentions his restoring a temple-tower at Babylon, implying that such a structure existed prior to his reign.

So now we have a good idea of when the building of Babel was started since Biblical chronology has Peleg living from 2269 to 2030 B.C.E.

When God scattered the people, was there anyone left at Babylon and what became of Nimrod after that?

It appears that Nimrod himself remained in the region after the first builders abandoned the tower. Note this from the Insight book:

"It appears that after the building of the Tower of Babel, Nimrod extended his domain to the territory of Assyria and there built "Nineveh and Rehoboth-Ir and Calah and Resen between Nineveh and Calah: this is the great city." (Gen. 10:11, 12; compare Mic 5:6.) Since Assyria evidently derived its name from Shem's son Asshur, Nimrod, as a grandson of Ham, must have invaded Shemite territory. So it would seem that Nimrod made the start in becoming a mighty one or hero, not only as a hunter of animals but also as a warrior, a man of aggression. (Gen. 10:8)

Observes the Cyclopaedia by M'Clintock and Strong:

'"That the mighty hunting was not confined to the chase is apparent from its close connection with the building of eight cities. . . . What Nimrod did in the chase as a hunter was the earlier token of what he achieved as a conqueror. For hunting and heroism were of old specially and naturally associated . . . The Assyrian monuments also picture many feats in hunting, and the word is often employed to denote campaigning. . . . The chase and the battle, which in the same country were connected so closely in aftertimes, may therefore be virtually associated or identified here. The meaning then will be, that Nimrod was the first after the flood to found a kingdom, to unite the fragments of scattered patriarchal rule, and consolidate them under himself as sole head and master; and all this in defiance of Jehovah, for it was the violent intrusion of Hamitic power into a Shemitic territory."—1894, Vol. VII, p. 109." (*** it-2 p. 503 Nimrod ***)
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And the city itself was rebuilt many times from that time forward:

"Nimrod, who lived in the latter part of the third millennium B.C.E., founded Babylon as the capital of man's first political empire. Construction of this city, however, suddenly came to a halt when confusion in communication occurred. (Gen. 11:9) Later generations of rebuilders came and went. Hammurabi enlarged the city, strengthened it, and made it the capital of the Babylonian Empire under Semitic rule." *** it-1 pp. 235-236 Babylon ***

Also see:

Does Genesis 10:5 really suggest that various tongues (languages) already existed BEFORE the confusion of tongues at Babel (Genesis 11:1)? (Search For Bible Truths)