The boy's parents are separated, with the doctor alleged to have carried out the circumcision for religious reasons before the infant was returned to his mother later the same day.
She complained to the police about the clinician claiming her son was in obvious distress when he was handed back to her.
Campaigners say it is believed to be the first time a police force has treated "non-theraputic" circumcision as GBH.
The 26-year-old mother says the child, now aged four, has suffered repeated physical problems, including inflammation and water infections, in the years since the operation as a direct result.
"It's even illegal to dock dogs' tails. I've come home crying my eyes out thinking a dog has got more rights than my child," she said.
"There's something seriously not right with it all. You can protect a dog, you can protect a girl, but not a boy."
The circumcision took place in July 2013. The mother, who lives in Nottingham, originally called social services, then contacted police on in November 2014.
There was an investigation, but Nottinghamshire Police initially decided it wasn't a criminal matter, and the case was referred to the General Medical Council.
However, the force reopened the inquiry after the mother got assistance from the anti-circumcision group Men Do Complain, and leading human rights lawyer Saimo Chahal QC, who wrote to senior officers.
Nottinghamshire Police said: "Officers investigating the circumstances surrounding a circumcision conducted on a child in July 2013 have arrested three people.
"A 61-year-old man was arrested on suspicion of grievous bodily harm with intent.
"A 44-year-old man and a 47-year-old woman were arrested on suspicion of conspiracy to commit grievous bodily harm.
"They have all been released pending further investigation.
"Our enquiries are ongoing."
