The Berne Convention was developed at the instigation of Victor Hugo of the Association Littéraire et Artistique Internationale. Thus it was influenced by the French "right of the author" (droit d'auteur) (...) Before the Berne Convention, national copyright laws usually only applied for works created within each country. So for example a work published in United Kingdom by a British national would be covered by copyright there, but could be copied and sold by anyone in France. (...) The Berne Convention followed in the footsteps of the Paris Convention for the Protection of Industrial Property of 1883, which in the same way had created a framework for international integration of the other types of intellectual property: patents, trademarks and industrial designs.
Have you heard of the British Empire? The UK did quite a lot to spread their legal traditions.
The UK made nothing to spread it, the US made sure it's more universaly accepted than the Human Rights convention.