First of all, DateTimePicker have nothing to do with DateTime they are two seperate things.
Second, the problem is in your method,you see you have a method(deliveryTime_ValueChanged) and when this method is called or triggered it'll run the code inside of it (deliveryTime.MinDate = DateTime.Now.AddHours(1);) and the problem is that there's nothing to trigger your method which is changing the date of your dateTimePicker, it's like you want to see what's inside of a potato chips bag without opening the bag.
i recommend that you use a timer tick event like so:
private void timer1_Tick(object sender, EventArgs e)
{
deliveryTime.Mindate = DateTime.Now.AddHours(1);
}
add a timer control and you can call it by Timer1.enabled = true;