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The highest fuse that can legally be used in a UK plug is 13A. That gives a max of 3,120 W, the US 15 A socket can supply 1,800W. So in the UK a 3kW space heater can be plugged into any socket.
Because the flex comes out the bottom, the wiring regulations (Electric code) requires a minimum height of 150mm above a horizontal surface.
In the UK the wiring regulations are mandatory everywhere; in the US the NEC (National Electrical Code) has to be adopted by each local city or county board. There are many counties where the NEC is not mandatory and anyone can do anything they want.
Unfused plugs rely on the circuit breaker in the breaker box to protect the circuit. In the US this is typically a 20 Amp beaker for a 15 amp circuit and 30 Amp breaker for a 20 Amp circuit. 20 Amp circuits are not commonly used in residential property. This means that the bedside light on that really thin wire is protected only by a 20 A breaker. The wire will melt and burn long before that. The UK plug is fused, the breaker is only protecting the permanent wiring. The lamp should have a 1A fuse which will blow to protect that really thin wire, long before the wire melts. 1, 3, 5, 7 and 13 A fuses are commonly available in the correct physical size for the plug,