All electrical systems must be protected by a fuse or a circuit breaker as a safety measure. These devices ensure that no more than a specified maximum amount of electric current is allowed to run through the system. Too much current is a sign of either a short circuit or too much load on the electrical system.
The size of the circuit breaker protecting the system determines the minimum allowable size of the wire used to feed the system. As wires increase in cross-sectional area, their resistance per unit length drops. Therefore, if a wire is too small for the current it must handle, it will have too much resistance and begin to heat up according to the equation P = I2R. The insulation around a wire can only handle so much heat before it begins to break down and no longer protect the wire. This can cause fires and short circuits, among other problems.
NEVER trim some of the strands off a stranded conductor wire to make it fit in a screw terminal that is too small for it. This derates the wire at the entry point to the terminal.
The minimum wire size for a given circuit breaker rating is strictly regulated by local laws and electrical codes to ensure a robust electrical system. It is important therefore that if a circuit breaker or a fuse is replaced with a larger overcurrent protection device, that the wiring being fed from it is upgraded as necessary. It is a serious violation to use a wire size smaller than that which can handle the maximum current allowed by the overcurrent protection device.
Why, then, in the United States, is it acceptable to design electrical appliances with such small power cords? Electrical outlets in the US are typically fed from 15A or 20A circuit breakers, and therefore have 14AWG or 12AWG wires powering them to handle the load. But clock radios, desk lamps, and other low current draw small appliances are designed with power cords rated for the expected current draw of the appliance, NOT the maximum current draw allowed by the circuit protection on the outlet! The key phrase here is "the expected current". If something goes wrong inside the appliance, it could easily experience unexpectedly high currents.
The United Kingdom understands the conflict at hand here and has taken appropriate action. Sizing every power cord for the circuit protection of the outlet would be cumbersome (especially in the UK where 32A circuit breakers are common). Electrical plugs in the UK are designed with a built-in fuse to limit the maximum current that can pass through the power cord. This way, the power cord can safely be sized for the fuse in the plug. This allows the power cords to remain small without running the risk of allowing too much current through too small of a wire.