LONDON—A Ukrainian lawyer named Olena Tyshchenko traveled to the Royal Courts of Justice on the afternoon of July 22 as a bit player in one of the most complex legal dramas ever seen here.

When she left the imposing edifice, though, she unwittingly walked into a starring role. Kazakhstan's state-owned BTA Bank had sued its former chairman, Mukhtar Ablyazov, accusing him of embezzlement and fraud on a monumental scale. It had won billions of dollars in damages.

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