My oldest son did this when he was this age, only with the bathtub. So, I filled the tub, sat with him in the bath and held him as we stayed in the tub while the water drained. He screamed, clung to me and was really scared. He was afraid of going down the drain with the water. As the water was draining, I just calmly told him, telling him- I'm here and we are not going down the drain. It worked. Once he realized that he wasn't going down the drain, he was fine and we never had any other issue with the tub.
So, something has clicked in your son's brain about the mower. Maybe he thinks he will be cut, just like the grass (totally rational!), just like my son thought he would go down the drain with the water.
I would take him outside when the lawn man is there. Before he starts the mower, let your son touch it. All over. Stand right there when the yard man starts the mower. Hug him, tall him that momma is not going to let him get hurt, only the grass is being cut. Follow behind the yard man as he mows. Yes, it will be very difficult, but stay the course. If you feel its too much, take him in and try again, from the beginning, the next time he comes to mow. This is called exposure therapy, if you want to google it.
I hope this helps, good luck!