Not starting on Key

We have an 08 reg C3 Stop and Start Sensodrive. It has a major problem. It has Stoped, and it will not Start!
It will not start on the key at all. Now we bought it in this condition and knew that it was a non-runner. It sounded like it should be an easy fix.
Boy were we wrong.
I will be as detailed as I can so that all you wonderful people out there in C3 land can come to an informed opinion and diagnosis. This vehicle has only done 29000 miles from new. It is well worth fixing.
We have a neighbour who was a good but untrained mechanic. He spent quite a while trying to find out what was wrong with it. He concluded that it was probably the Starter Motor. I agreed.
We then found that the Stop and Start Sensodrive does not have a starter motor. I expect all you knowledgeable people know that.
The Starter Motor is incorporated into the Alternator. It is expensive. Very expensive.
Well, we bit the bullet and bought one. £720.00 exchange, if you want to know.
It wasn’t that. It still won’t go.
Now a mobile mechanic had taken over the refitting of the Alternator because the original person that was doing the job had a major stroke.
When the mobile bloke had done the refitting and it didn’t fix the problem, he gave up but had an idea that it could be the sensor in the gear shift. I am pretty sure it is not that for reasons I will come to in a moment.
So what does happen when you turn the key?
Well, the first thing that happens in a “BEEP” is heard. I don’t know what that is for.
The display in front of the steering wheel shows as can be seen in the photographs. The battery light is flashing as I believe it should do. It shows the vehicle is in Neutral and the picture with AUTO and R show that it will move into the gears and the vehicle does know what gear it is in. So it is not the sensor. I think.
The middle display shows, “INFO – ECONOMY MODE – ACTIVE”
Shortly after this, the displays will go out and if you turn the key off and then back on, the displays will not re-light. In all of this, I have my foot on the brake.
If I remove the key, wait a few seconds, put my foot back on the brake and turn the key on. The displays will come on again for a short while.
Interestingly, if a battery boost unit is attached to the battery when the key is turned, the radio will come on. It does not do that before. It is a brand-new battery.
So there you have it. Has anyone any idea what the problem is? Is it a simple fix that an idiot (me) can do? Please don’t confuse me with jargon, I am new to this!
Thanks,
Phill

The car will automatically go into economy mode if the ignition has been on and the engine not running for longer than 30 minutes. That's perfectly normal. You can not get the car out of economy mode until the engine starts. So don't worry about that.
There are a number of components in the "stop and start" system that can stop it from working. The alternator/starter is only one of them. There is the stop start module in the engine bay and a small auxiliary battery under the passenger seat. Not to mention any wiring or connectors in between.
I would highly recommend taking the car to your local dealer and get the fault codes read. Just get the codes. Don't get it diagnosed or have any other work done, just get the codes!
Without the codes you will be fumbling around in the dark.
It does not appear to be the sesnsodrive as it changes gears and there is no "snowflake of death". If there is a problem with the gearbox a snowflake symbol will pop up and flash.

Thank you for that.
Just as an addition to my post.
The battery under the seat is fully charged.
We know the engine runs as when it was brought to me on the back of a truck, as we ran it off the flatbed we tried bump starting it and it ran well.

Yes that pretty well confirms the problem is in the stop start system. But without some sort of clue you would be replacing parts willy-nilly.
That could prove expensive.
There are a whole bunch of inputs to the stop start system. For example;
The drivers door switch.
Reverse switch.
Drivers seat belt switch.
The outside temperature sensor.
Battery temperature sensor.
Brake booster pressure switch
Stop start button on the dash.
Brake switch.
The stop start control module sits to one side of the front of the radiator.
Could be any of these!?
The only thing that we do know is that it's not the alternator/starter!

Connect it to lexia tool to interrogate the start-stop system for recorded fault and potentially, live data.