In recent tournaments, nations can't play Olympic matches without both FIFA & IOC memberships How about past tournaments ?
In 1956 and 1965, a unified Germany team played in the Olympics exclusively. And as late as 1960, a Great Britain team was in the Olympics. Even in 1908, Bohemia was forced to withdraw because they lost FIFA membership, although in that same tourney, France entered two squads (A and B team). The first two tourneys in 1900 and 1904 featured club squads, not national teams.
I seem to recall some noise being made that the UK would be submitting a combined team for the London 2012 Olympics, since they have host qualification - don't know what happened with that idea.
IIRC the non-English UK nations don't want anything to do with it. The compromise they appear to have reached is that the UK team will just be an English team in all but name. This for both men and women. J