I want to be understanding of you needing to talk to your kids and applaud you put them first.
But a few things to keep in mind.
If you scheduled for say 2 pm do you expect me to bet here at 2? Is it ok for me to be late?
No? Then if you expect me to be on time reverse should be true. You know someone is expected to arrive at a given time so you should call the kids and yell at them prior to the appointment and be ready when the time comes. What kind of session is the guy going to get if just before walking in the door you have been fighting with the kids and are angry? Can you really turn that off and be pleasant so the planned activities will be top notch?
Remember whatever amount of time you ask the guy to wait he is sitting in the car looking suspicious. When people see him there do they think maybe he is casing the joint and up to no good? If a cop pulled up and questioned him how does he explain just sitting around in a car at a place he has no good reason to be? Do you expect him to say he is waiting for you to give the ok to come up so yes he has a good reason to be there?
And if it is summer remember he is sitting there in a hot car getting sweaty. He is going to be sweaty smelly stinking when he walks in. If personal hygiene is not important to you then no biggie. If you make it a point to be freshly showered (as you should) and expect the client to be freshly showered (as you have a right to) then even one minute is too long. In 100 degree heat it does not take long. I remember having an appointment with someone once and arriving to hear them say they had not arrived yet and would be there shortly. I waited around 10 minutes and I could stand to smell myself. I was miserable from sitting out in the heat and any romantic mood I had been in was way gone.
If your appointment time is going to be a moving target discuss up front and let the person know. If something unexpected comes up when the guy calls and says I am here give him a definite time of what the delay is expected to be. Then if he needs to drive around the block some or go get an ice cream at Braums or whatever to stay clean and comfortable you are less likely to have someone give up and move on to plan B.