Commonwealth of Independent States (CIS) circa 1992 for a number of sports (and under a number of different names), although that's more of a slowly splitting up into multiple nations.
If it counts, Serbia and Montenegro played the 2006 FIFA World Cup as one team shortly after both countries had declared independence.
Ireland play as All-Ireland (Republic of Ireland + Northern Ireland) in a number of sports, most notably rugby and cricket, and then in the Olympics there's a strange situation where Northern Irish athletes are able to choose between Ireland and Great Britain (Team GB itself is a case of athletes from Great Britain as well as its territories around the world who are usually independent in sports terms).
Another from cricket is some of the teams that enter as regional selections like the West Indies (is 'East Africa' still a team?).
As for Team GB's football, too late now for 2016, but the powers that be really should try and work something out. I've said it before that this issue should be settled on the pitch. If one nation does well enough to earn a spot, let their FA control the team. If more than one qualifies, either work together like grown-ups or just go home