Using a TemporalAdjuster to round a time to the actual nearest minute, i.e. rounding to floor or ceiling as appropriate. While truncation (rounding to floor) is part of the standard Java API and also implicitly done when formatting only hours and minutes, rounding up or down to the nearest minute is not.

NearestMinuteTest.java

import static org.junit.jupiter.api.Assertions.assertEquals;

import java.time.LocalTime;
import java.time.temporal.*;
import org.junit.jupiter.api.Test;

class NearestMinuteTest {
  
  /**
   * Adjusts given Temporal object to the <em>nearest</em> minute (rounding up from 30 seconds),
   * assuming it has the necessary field (SECOND_OF_MINUTE). This includes at least Local(Date)Time, 
   * Offset(Date)Time, Zoned(Date)Time.
   */
  public static final TemporalAdjuster NEAREST_MINUTE =
      (t) -> {
        if (t.get(ChronoField.SECOND_OF_MINUTE) >= 30) {
          t = t.plus(1, ChronoUnit.MINUTES);
        }
        return t.with(ChronoField.SECOND_OF_MINUTE, 0).with(ChronoField.NANO_OF_SECOND, 0);
      };


  @Test
  public void tests() {
    LocalTime lt = LocalTime.of(13, 59, 30, 0);
    assertEquals(LocalTime.of(14, 0, 0), lt.with(NEAREST_MINUTE));

    lt = LocalTime.of(13, 59, 30, 1);
    assertEquals(LocalTime.of(14, 0, 0), lt.with(NEAREST_MINUTE));

    lt = LocalTime.of(13, 59, 29, 999_999_999);
    assertEquals(LocalTime.of(13, 59, 0), lt.with(NEAREST_MINUTE));
  }
}