Malaysian businessman Fernandes had been with the club since 2011, though now says "It's time for the club to have a chairman who is based in London."
Bhatia, who joined the QPR board in 2007, will swap roles with Gnanalingam and work alongside chief executive Lee Hoos.
Fernandes also remarked: "It's a good time for a change as the club is now in a stable position and this will give the new chairman a good foundation to build upon," the 54-year-old said.
"It was important for me to step down at a time where we have solved many issues such as Financial Fair Play (FFP), making the club sustainable, building a strong academy and having a strong backroom team."
The 'Hoops' were fined £17m in July as part of a £42m settlement for breaching FFP rules.