Russia's invasion of Ukraine meant the original plan to play Scotland in March was shelved and the knock-on effect was that Wales has also been kept waiting.
To reach the finals in Qatar, Ukraine must win away to Scotland on Thursday 2 June (AEST) before coming out on top in another away game against Wales on 6 June (AEST).
Wales secured its place in the final round of the play-offs with a 2-1 victory against Austria on 25 March (AEDT), which had been the original date for the Scotland-Ukraine fixture.
The winner of the final European play-off will go into a FIFA World Cup group alongside England, Iran and United States.
UEFA announced the new play-off dates on Friday (AEST), as well as explaining how its UEFA Nations League opening fixtures, also set for the June international window, would be rearranged to allow for Ukraine, Wales and Scotland to fulfil their FIFA World Cup commitments.
European football's governing body said it had held "extensive discussions" with eight national associations who would be affected by the FIFA World Cup games being switched to the intended UEFA Nations League dates and said there had been "a remarkable spirit of solidarity and cooperation" in negotiating an updated schedule.
Wales was due to play Poland in the UEFA Nations League on 4 June (AEST), but that game has been switched to 2 June (AEST), meaning Rob Page's team will have a competitive game in the build-up to facing either Scotland or Ukraine.
Scotland had been due to play Ukraine in the UEFA Nations League on 8 June (AEST), but that game has been shunted back to 22 September (AEST), with both teams then due to play three games in seven days.
Nations League group A4 and B1 games affected by FIFA World Cup play-offs, with new dates (all dates AEST):
2 June 1: A4 - Poland v Wales
5 June: B1 - Armenia v Republic of Ireland
9 June: A4 - Belgium v Poland, Wales v Netherlands; Scotland v Armenia, Republic of Ireland v Ukraine
12 June: A4 - Netherlands v Poland, Wales v Belgium
15 June: A4 - Netherlands v Wales, Poland v Belgium
22 September: B1 - Scotland v Ukraine
25 September: B1 - Scotland v Republic of Ireland, Armenia v Ukraine
28 September: B1 - Republic of Ireland v Armenia, Ukraine v Scotland