"They had a strategy to deal with a ‘death of Stalin’-type scenario," Johnson said.
Nicholas was a nod to Nick Price and Nick Hart - two doctors who the couple have praised for saving Johnson's life at St Thomas' hospital last month.
Both mother and child are doing well. The boy was born at a London hospital.
The United Kingdom has the fifth worst official death toll in the world, after the United States, Italy, Spain and France.
"On the advice of his medical team, the PM will not be immediately returning to work. He wishes to thank everybody at St Thomas' for the brilliant care he has received," an official statement said.
Downing Street said Johnson "continues to make very good progress," but interior minister Priti Patel said it was vital he took time to fully recover.
Hedley faced accusations of antisemitism last year for telling a British activist who combats Jew hatred, "You are a modern-day Nazi.”
"He is in extremely good spirits."