Lockyer, 75, was sentenced on Friday at Woolwich Crown Court after being found guilty at the same court on November 26.
Lockyer was jailed for a total of two years for causing death by careless driving and six months to run concurrently for causing death while unlicensed.
The 75-year-old was also disqualified from driving for two years.
During the trial, the court heard Lockyer did not slow down or brake when he saw the first two boys crossing the road in front of his car.
He maintained his speed and Josh was partially obstructed from the pensioner’s view due to the geography of the road.
Lockyer’s brake lights illuminated 0.4 seconds before he collided with Josh. He initially slowed to about 35mph after the impact but then accelerated away, back up to 45mph.
Another motorist went after Lockyer and confronted him at the next set of traffic lights. The motorist tried to block the pensioner in but Lockyer reversed and drove around him.
He eventually returned to the scene after doing a U-turn at the next junction. He was arrested and taken into custody.
Scotland Yard said its investigation revealed Lockyer had failed to renew his driving licence after it had expired.
Despite being a MOT examiner, he had not had a valid driving licence since January, 2019.
Detective Sergeant Steve Cannings, of the Met's Serious Collision Investigation Unit, said: "This is an absolutely tragic case in which the life of a young boy has been lost due to an act of careless driving.
"The anguish of Josh's family has been compounded by the fact Lockyer should not have been driving at the time of the collision as he did not have a valid licence.
"I would like to praise the dignity and courage of Josh's family who have attended every day of the trial and I support their efforts to ensure a case like this never happens again."