Toto Wolff has also provided a further update on the team president’s health following the surgery.
Lauda, 69, picked up the flu while on holiday in Ibiza but the problem soon developed into pneumonia and then “severe lung disease.”
A new pair of lungs were flown in from Germany and the operation went ahead successfully on Thursday.
Lauda was treated in his native Austria and is now on a long road to recovery – and he has been wished well by F1 star Hamilton.
“Send you all my love, prayers and support, Niki,” Hamilton wrote on Instagram.
“Wishing you a speedy recovery.”
Mercedes boss Wolff underlined that Lauda is in no rush to return to work following his illness.
“The world knows Niki as an F1 legend with incredible power and resilience,” Wolff said.
“For all of us at Mercedes he is our chairman, our mentor and our friend.
“We have missed him by our side in Hockenheim and Hungary and can’t wait to have him back with us on the grand prix tracks of this world.
“The recovery he now faces is not a race. But knowing Niki, I’m sure he will soon be telling every nurse and doctor that he has had enough of the hospital.”
Vienna General Hospital are still waiting to see how Lauda’s body reacts to the new lungs.
And they released a statement following the operation which read: “Due to a severe affection of the lungs, Niki Lauda had to undergo a lung transplantation at the AKH Vienna today.
“The transplantation was done successfully by Walter Klepetko, head of thoracic surgery, and Konrad Hotzenecker.
“We kindly ask for your understanding that the family won’t give any official statements and we ask to protect the privacy of the Lauda family.”