In the 19th century, Britain was by far the richest country in the world, and the wealth/profits reaped from global colonies ended up in the UK and were invested in science, engineering, arts etc...
TV, radio, internet, computer, jet engine, steam engine, iron ships, printing press, the list is endless.
Brits don't claim to be the first to kick a 'soccer ball/football', per se, but the rules of the game as we know it today were invented at Eton College London.
Brits also invented hockey, cricket, golf, rugby, snooker, tennis, this list is also endless.
The British Empire simply spread these British inventions and luxuries across the globe - so I conclude that, if you were to single out the nation who has influenced the modern world the most, it's the British, by miles.
However, people will claim it is us (the USA), and to some extent we have taken over where the British left off. We fight wars using tanks and aircraft (both British inventions), we made movies in their language and using TV, radio and the internet (all British inventions).
In essence, the British made the modern world and the USA milked it.