Harry Maceachen, from Shrewsbury, was born with a rare disease and had the life-saving transplant on Thursday.
His father Simon donated part of his liver after no suitable match was found from the organ donor register.
Grandmother Alison Price, a former theatre sister, said: "The consultants are very happy with their progress."
She said it had been a "very, very difficult 12 months", adding: "The operations were satisfactory. Everything is going on according to plan....the consultants are very happy with their progress."
Harry was born with biliary atresia, which meant he had blocked bile ducts.
He had a transplant before his first birthday but that liver had begun to fail, so a second one was required.
Harry underwent the operation on Thursday at Birmingham Children's Hospital and Mr Maceachen was operated on at Queen Elizabeth Hospital in the city.
His grandmother said: "I used to look after him (Harry) three days a week when his mother went back to teaching and he got to know me very well.
"I think we've got a special bond. He was born on my 63rd birthday."
She said Harry's two-year-old brother, Sam, has been "impeccably behaved".
"It's hard for him. It's the first time he's been away from his mother for as long as this and he obviously misses Harry, as we all do."
She added: "It's brilliant with the support (the family have) had and the support we've had as grandparents.
"The whole family's pulled together and they have a wonderful lot of friends."

A 38-year-old man, who was a back seat car passenger, is in a critical but stable condition in hospital following the B9080 crash at about 22:15 on Wednesday.
The injured passenger suffered serious stomach and abdomen injuries.
The Vauxhall Zafira was travelling south when it crashed with a First bus between Kirkliston and Winchburgh.
The driver of the car, a 58-year-old man, suffered spinal injuries and a fractured ankle.
The front seat passenger, a 34-year-old man, had injuries to his face and leg. All were admitted to hospital.
Three of the bus passengers were treated in hospital for minor injuries. They have all since been discharged.
Police said the driver and six other passengers on the single-decker bus were treated at the scene.
The bus had 26 people on board.
A First Bluebird spokeswoman: "We can confirm that one of our Service 38 vehicles was involved in a collision yesterday at around 22:15 between Kirkliston and Winchburgh. The bus was travelling to Falkirk from Edinburgh.
"Unfortunately the incident resulted in a number of injuries and our thoughts are with the injured at this time.
"We have launched an investigation to establish exactly what happened and are assisting Police Scotland with their inquiries."

The Highland village's Ruthven Road is mentioned in a line in the film delivered by Janet de Vigne, from Edinburgh.
De Vigne's appearance at the screening will mark the launch of new weekly film nights in Kingussie.
The village does not have a cinema, but will hold the events in a local hotel.
Holding weekly film nights was suggested during an annual festival celebrating movies.
Organiser Iona Malcolm said: "This was all born at the Kingussie Food on Film Festival when a survey was carried out into interest in a film night in Kingussie.
"The reaction was very, very positive."
Directed by Danny Boyle, Slumdog Millionaire won eight Oscars in 2009.

A 20-year-old man was with a male friend in Old Rutherglen Road when another man attacked him with a weapon at about 13:05 on 22 September.
The victim was treated in hospital for a leg injury.
Detectives believe the man in the images may be able to help with their inquiry. He is described as white, aged about 20, with short brown hair.
At the time of the incident, he was wearing a tracksuit top that was white at the front and light blue at the back, dark tracksuit trousers and dark trainers.
He was last seen walking on Caledonia Road towards Cathcart Road.
Det Con Nicol McPherson said: "This was a violent attack that took place in broad daylight which left this man with a serious leg injury.
"I would urge anyone who recognises the man in the images, or who has information that may assist the investigation to contact Cathcart Police Office."
The grandmother of a four-year-old boy who received part of his father's liver has said all is going "according to plan" following the operation.
[NEXT_CONCEPT]
Police are appealing for witnesses following a head-on crash between a car and a bus in West Lothian.
[NEXT_CONCEPT]
An actress who appeared in the Oscar-winning film Slumdog Millionaire is to attend a special screening of the movie in Kingussie later.
[NEXT_CONCEPT]
Police have released CCTV images of a man they want to trace over a "violent attack" in the Gorbals area of Glasgow.