The 18-year-old, who won bronze at the Rio Olympics in 2016, will begin her rehabilitation under the guidance of the British Gymnastics medical team and English Institute of Sport support staff.
"I'm absolutely gutted and heartbroken right now. My dream was to go to the Commonwealth Games and compete for my country but I will come back stronger," Tinkler said in a statement released by British Gymnastics.
"I will be working hard on my rehab and the aim, if all goes well, is to put myself in contention for the British team for the European Championships in Glasgow this summer."
The Commonwealth Games will be staged from April 4-15 on the Gold Coast, Queensland.
(This version of the story clarifies Tinkler would have represented Team England at Games.)
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
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