Their wiring is different, they don't use the heatlink for one, the Nest controller does it all, hence the 10 terminals rather than the 2 in the UK model. I guess it would be difficult to get power to the unit without extra wiring.
Not using electric when a gas supply is available would be a start! Switching to electric will be incredibly expensive to run. We look after a few houses with such an arrangement, requiring large stores of hot water for the heating, and hot water, heated with 18KW worth of immersion heaters...
The refill time on mine takes a while too, though not that long, but I have high water pressure. If it has suddenly lost all that water there may be a leak, or the expansion vessel and pressure relief valve may require servicing/replacement. Also check the pressure relief pipe that goes outside...
The boiler comes with a key that fits these 2 valves. Turn both 90° anticlockwise to fill boiler, then turn both back. The slot indicates the valve position, front to back closed, left to right open. They are often stiff if they haven't been operated in a long time. Some radiator bleed keys will...
The heatlink (what the nest controller talks to) has 2 terminals T1 and T2, these connect to the 2 terminals on the wall mounting plate of the nest controller to provide power without the usb.
Wow what an absolute mess. Not to mention using the bare earth wire as a switched live. Terrible. I would recommend contacting someone competent to work on heating wiring to install it properly, and safely.
The capacitor can age with use, where its capacitance value drops. A very common problem in all sorts of things that use film capacitors in mains positions.