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This sounds again like a fault I encountered a few days ago with another transit.
This could be one of a couple of faults.
As you say that the batteries under the seat are both good and charged that eliminates my first fault.
OK. As I don't know your level of expertise I will try and make it simple.
The first fault it could be is an earth fault to the starter motor. With a multimeter check the voltage across the starter with the red probe on the main positive on the starter and the black probe on the body of the engine. This should read 12 volts. If it is less then this points to the earth on the starter and what usually needs to be done is undo it, clean it up and refit it and then check it again. If All is good try and start it and it should be OK.
Another thing to check is the leads to the starter itself and also the connections. What would be a good idea is if you cleaned these as well as these can also corrode. But when removing the main power from the starter you will have to remove the leads oft of the battery so as not to cause a spark if it touches the engine.
If these faults have been eliminated then one other fault could be the ignition lock itself as they do sometimes fail but this is very rare.
One other thing it could be the starter itself could be jammed. They used to suffer with this fault a lot as the clutch dust gets into the starter and clogs it up making it jam but they usually give a bit of trouble first before this happens.
I would always check the earth's first and make sure they are all good and clean before looking at the starter etc.
I hope this had helped you and if you require further assistance please don't hesitate to ask.
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Thanks in advance