Bairstow and Ali have over 400 international appearances between them but the veteran pair have been deemed expendable as England revamp their limited overs line-up following disappointing ODI and T20 World Cup campaigns.
Matthew Mott was sacked as England's limited over head coach last month and the changes continued with Monday's squad announcement.
Five uncapped players have been called up for Marcus Trescothick's first assignment as interim coach.
Left-arm seamer Josh Hull, all-rounder Jacob Bethell and pace bowler John Turner were selected in both formats, while Dan Mousley and Jordan Cox come into the T20 group.
Moeen had been an influential vice-captain to Jos Buttler but is now 37 and has acknowledged his international career was winding down.
His role as a spin bowling all-rounder will be covered by his Warwickshire team-mates Bethell and Mousley.
Bairstow turns 35 next month but there is no such sense that he is ready to bring the curtain down on his England days.
He was an integral part of the white-ball revolution that carried the side to 50-over World Cup glory in 2019, scoring centuries against India and New Zealand in must-win games at that tournament.
Despite having around 14 months left on the two-year central contract he signed last October, Bairstow lost his Test spot earlier this year after winning his 100th cap over the winter.
Chris Jordan has also been axed, while Liam Livingstone has retained his T20 spot but misses out on the 50-over matches.
Durham seamer Brydon Carse was selected after completing a three-month ban for historical betting offences on Wednesday.
He has not played competitively since May 10 due to his suspension but has been offered an instant return by England's selectors.
Jofra Archer, whose only cricket since the T20 World Cup has been in the Hundred, retains his place in both squads.
The three-match T20 series against Australia starts at the Ageas Bowl, Southampton the day after England's third and final Test concludes against Sri Lanka.
Harry Brook, Gus Atkinson, Ben Duckett, Jamie Smith and Matthew Potts will take a short break before joining up with Trescothick's ODI team for the five-match series.
Joe Root is not included in what is believed to be a case of workload management ahead of a busy winter.
Zak Crawley, Ben Stokes and Mark Wood are all out injured but fast bowler Saqib Mahmood returns after recovering from his own fitness issues.
England T20 squad to face Australia: Jos Butter (capt), Jofra Archer, Jacob Bethell, Brydon Carse, Jordan Cox, Sam Curran, Josh Hull, Will Jacks, Liam Livingstone, Saqib Mahmood, Dan Mousley, Adil Rashid, Phil Salt, Reece Topley, John Turner.
England ODI squad to face Australia: Jos Buttler (capt), Jofra Archer, Gus Atkinson, Jacob Bethell, Harry Brook, Brydon Carse, Ben Duckett, Josh Hull, Will Jacks, Matthew Potts, Adil Rashid, Phil Salt, Jamie Smith, Reece Topley, John Turner.
