The England and Wales Cricket Board (ECB) announced Wednesday that Cooley, a 55-year-old Australian, will aid the tourists' quicks in Brisbane under the direction of elite pace bowling coach Jon Lewis.
Following Rafiq's comments, which have sparked a tidal wave of accusations of racism within English cricket, the 47-year-old Vaughan was stood down from his BBC radio show earlier this month.
Harrison faced fresh criticism for his response to revelations of racism by former Yorkshire player Azeem Rafiq after they both spoke in front of a parliamentary committee this week.
The accusation against Gale does not refer to the Rafiq case, but a tweet he sent in 2010.
England and Wales Cricket Board (ECB) chief executive Tom Harrison met Raja and the pair agreed to reschedule the men's matches.
According to sources in the PCB, besides meeting Raja, Harrison is also due to call Prime Minister Imran Khan in Islamabad
The 30-year-old Rafiq accused Yorkshire of failing to deal adequately with his allegations of racism during his time with the English county.
The cricket club recently concluded that no employees would face any action over Rafiq's claims of institutional racism, leading to major sponsors this week abandoning deals over the handling of the case.
Rafiq made serious allegations about his experiences at the club and Yorkshire appointed lawyers to launch an investigation, with an independent panel also put in place to oversee it.
The Edgbaston Test will be part of India's white-ball tour of England. The match will be held from 1 July.
Following a breakthrough in talks with Cricket Australia, the England and Wales Cricket Board on Friday said they had given "conditional approval" for the tour to go ahead.
The ECB has also decided to do away with their old process of announcing separate red and white-ball contracts, announcing instead a single contracts list for players across formats.
Jos Buttler could be the only England player not to agree to travel to Australia.
England's men's and women's white ball teams were due to tour this month but the England and Wales Cricket Board (ECB) pulled out of both because of concerns over "mental and physical well-being".
Ahmed, who has a foot in both camps after serving as England's spin-bowling coach in 2009-2015, voiced his frustration to AFP after English cricket chiefs controversially cancelled tours for their men's and women's teams next month to cricket-crazy Pakistan.
Moeen has scored 2,914 runs in 64 Tests at an average of 28.29 with five centuries and picked up 195 wickets with his off-spin bowling at an average of 36.66.
The fifth and last Test between England and India was cancelled after COVID-19 spread in the Indian dressing room, leading to most of the support staff catching the virus and going into mandatory quarantine.
New Zealand abandoned and England pulled out of scheduled assignments in Pakistan in quick succession citing security concerns. England also gave players' mental health as a reason for the withdrawal.
The ECB on Monday withdrew its men's and women's teams from the tour of Pakistan, citing increased concerns about travelling to the region.