Ford Repair Questions? Ask a Mechanic for Answers ASAP
It sounds from your description that one of the Starter Relays is breaking down when hot. As it cools down it then starts working again hence why it will once again start.
You have stated that you have had the Starter Motor replaced. Was the Starter Motor Count reset at the same time? This is a vital part of replacing the Starter Motor on these syatems.
I suspect it is the Start / Stop relay. This is constantly working whenever the vehicle is obviously starting and stopping. Considerable heat can build up here and the way you describe it starting fine when it cools down indicates that it is something in this realm. The only other part I would check is the relay that powers the Engine Control Module as this too can exhibit the same issue.
Starter Motor Count is the Count that the ECM registers every time the Starter Motor is used. When it get s to X amount on the Count then it will fail to start until the Starter Motor is replaced and the Count reset.
Unfortunately it appears you have used the wrong workshop for this work. A Ford Dealer will have the required diagnostic software, Ford Trained Technicians who know the product and Ford Technical Information at their fingertips.
It is more expensive going to a dealer in terms of Labour charges however you are paying for people who continually go on training courses to keep them up to date with the latest information about your vehicle. They also fit parts that are made for your vehicle.
You might say £650 is expensive but lets think about this. You've already paid £350 to these people and the vehicle isn't fixed. The next stage will be them throwing something else at the vehicle in the vague hope that eventually it cures it! They play a system that I call Parts Darts! Keep fitting things until it is fixed! Might end up costing you £1500 due to their lack of knowledge!