In the olympics England Scotland and Wales are not separated. More strangely, it's a team GB that includes Northern Ireland, but they refused to call it team UK...
When England and Scotland played their first football game in 1872, there were no real intent to expand international events beyond these 2 countries. IFAB was founded in 1886, mainly to organize the British Home Championship that started out in 1884, but still with no ambition to expand beyond the British Isles. In rugby, the "Home Nations" which is the ancestor of today's "6 Nations", appeared at the same time, a bit earlier in 1882. On the other hand, the IOC always had the ambition to make of the modern Olympic games a truly international event, in which all recognized countries in the world should participate. Hence why it is "Great Britain" which joined, and not its constituent countries.
In the early 1900's, football was developing fast in continental Europe but in a totally disorganized way. Dutch banker Carl Anton Hirschman tried first to join new European federations to IFAB, but IFAB wasn't interested. Therefore he proposed to found FIFA between continental countries in Paris in 1904. For its own credibility, FIFA needed British nations to participate considering the sport was already well developped there, so they accepted constituent countries, already members of IFAB, to join as their own entities. England will eventually join FIFA in 1905, negociating that the English FA President would lead it, but more as a way to control it than to really develop it. England will leave FIFA as a pretty ghost entity in 1920, and Jules Rimet will get it back in 1924 with the objective to create a world cup that will give it a real purpose, which will be done in 1930. Yet England will only recognize it later in 1946 when it will rejoin.
As for the name of the country, it is officially "the United Kingdom of Great Britain and Northern Ireland", shortened as "Great Britain". It's only after the end of the British Empire that it started being shortened as "the UK", mirroring "the US" to designate "the United States of America". For what it's worth, "England" was still using the Union Jack as its flag in football in the 1980's, the Saint George's cross flag only replacing it during the 1990's.