The incoming mains supply at my house is 6mm copper cable, which is rated at 47A - I believe more modern properties get 10mm cable?, rated at 65A.
Sockets are wired using 2.5mm cable (27A) in a ring, so each socket is fed from two pieces of 2.5mm cable, but this same ring usually includes all sockets on that floor. Each plug that connects to the sockets also includes it's own fuse.
Lights are fed using 1mm (14.5A) to 1.5mm (20A) cable, usually each floor will have it's own fuse in the fuse box, or perhaps more.
Cookers are hardwired using either 6mm cable, or 10mm cable - fed from it's own fuse in the fusebox, likewise for a shower.
Aircon is pretty rare here, but is normally hardwired to it's own fuse, with cabling based on it's requirements.
Each house is normally supplied one phase (out of three) and a common neutral - houses are connected alternately to different phases to spread the load. Commercial properties often have the full three phases supplied, so can run much higher powers.
We don't get the cascading failures over here, the infrastructure prevents that.