For example, the US was instrumental in enacting the UN Single Convention on Narcotic Drugs, so a government that I don't get to vote for had a direct influence on the laws of my country (UK).
There is some distance between the US and the UK but it's still comparatively modest compared to the world as a whole.
If, for example, Asian and Middle Eastern countries influenced the law of this hypothetical world government in proportion to their population, then the UK might be compelled to execute violators of the drug laws you refer to.
If, for example, Asian and Middle Eastern countries influenced the law of this hypothetical world government in proportion to their population, then the UK might be compelled to execute violators of the drug laws you refer to.