I think it is most properly represented as a mix of both.
Prior to WWII, British (UK) English did expand because:
It was the language of administration of the Empire. If the natives wanted to be anywhere near the seat of power, they had to learn English. If they could afford to, many would send their children to study in England1.
Commercially, you would access to British goods through British firms, so you would need to have someone to translate the correspondence with them. And the UK was the model of a successful nation that everybody tried to imitate.
That said, trade before WWII was relatively limited (remember, ships were loaded and unloaded by hand), communication media few and slow and, for most countries, literacy rates even in their own language were quite low, so very few people would have been English speaking.
After WWII, the UK loses the Empire and the USA takes its role as a commercial and manufacturing powerhouse, and we see the development of:
- Mass transportation, which causes both an increase in trade and other exchanges (e.g. tourism, migration) 2.
- Massive presence of US military bases around the world, exposing people who otherwise had no foreign contact to US English.
- Continuous improvement of school systems, which last for more years and -often- include foreign language studies.
- Explosion of mass media, which gives the public access to culture in English3. In some cases, even local artists begin using English because of the "cool" factor.
- Add to that the fact that in most countries there was already a small nucleus of English-speaking people who could help teach other people, which would have accelerated the change.
Note that I am not saying that all of the expansion of English post WWII was due to the USA4. I am pointing that the factors that favored the expansion of English appeared in a timeframe where the major English power was the USA, but the UK still had some influence at that time (through tourism, trade, culture, etc.).
1 In fact, the same happened in France colonies, but as stated somewhere those accounted for less population.
2 Ok, in some places like Europe, English-speaking tourism was principally from the UK and not from the USA.
3 Even something as "silly" as understanding pop song lyrics was a motivation to learning English that our parents had but our grandfathers did not.
4 And, conversely, probably there were parts of the world prior to WWII where English was introduced due to US influence.