Sounds like... F125!
This is the same problem Neil had when the neutral lockout terminal fell off inside the switch - there is a relay in the starter solenoid circuit which is only activated in P or N, otherwise disconnects the solenoid.
It can be bypassed by a bit of chunky wire and some 6.3mm flat terminals if you're unable to get it to a garage to swap the F125 - basically the same configuration as a manual car. There might be a pic in my manual swap thread.
Don't throw the old switch away as I can usually refurbish them if its just a burned or bad contact (like most of them). Not so much if the contact has actually come adrift though like Neil's did.
Whoever designed that stupid switch has a lot to answer for
