Can you test each zone by itself?
I would try that and see if you can run atleast one, then turn the other zones on one by one until (if it happes) you loose water pressure.
If you can't run one zone, check it very carefully for cracks, disconnected pipes, ect. If that check out then your
water supply (city or well) can't handle the load and you might have to get an extra water tank + pump.