Fiat 500 2010 won't start but is ticking over and will only open with key fob some times and RAC man has no idea what won't this is his report
Actually I'd say the RAC report is a pretty good summary of what's likely to have happened here.
I'm guessing your car has S/S fitted (it became standard from the 2010 model year).
If so, be aware that connecting any form of charging or jump starting equipment directly across the battery terminals can permanently damage the battery management and charging electronics. There is a specific warning against doing so in the documentation that came with the car. IIRC when I collected my car from the supplying dealer, I had to sign a form saying that I understood this before taking delivery. There were a few different variants of the system fitted to the early cars and some may be more sensitive to damage than others.
If the RAC report is correct, you've tried to recover a flat battery by connecting a charger directly to the battery terminals and have burned out the charging electronics. You'll need manufacturer-specific software tools to diagnose this properly (most independent garages and auto electricians won't have this), so you're probably going to have to get the car to either a franchised dealer or an independent FIAT specialist. It might be possible to diagnose using
multiecuscan.
Whatever you do, you're almost certainly going to need a new battery. If you go down the franchised dealer route, you might first want to buy one from any of the usual sources, as you'll likely save £100-£150 on the battery cost. You must fit a battery specifically designed for S/S use; it'll cost you a little under £100 if you buy it from one of the discounters & fit it yourself; Fiat dealers typically charge £200-£250 fitted.
Sadly I don't think this is going to be cheap; if the electronics are damaged, this will be considerably more than the battery cost. Please ask if you have any further questions and do let us know how you get on.
For anyone else with a S/S car, please take this as a warning as to what may happen if you try to recharge a battery without understanding the specific requirements of Fiat's S/S system, and never, ever, connect anything directly across the battery terminals. If you must recharge or jump start with the battery connected, connect the +ve lead to the + battery terminal, and the -ve lead to a secure earthing point on the chassis.