I have just done a very quick and very unscientific poll* of the effect of Scotland gaining independence and the impact this would have on Great Britain on the Olympic medal table, as it currently stands.
The good news is the United Kingdom (without any Scottish athletes) would still be in third place in the table, with 11 gold medals. However, this is only 1 more than Korea.
Scotland on the other hand would definitely have 1 gold medal (Andy Murray), placing them joint 31 with Croatia and Switzerland because they also achieved 1 silver (Michael Jamieson)
There are of course a number of other factors that need to be taken into account. As a majority of the money comes from the National Lottery would Scotland be able to support their athletes to the same standard as the rest of the UK? Also would the the rest of the UK be able to replace the Scottish members in the team golds with equally as good athletes meaning they would rack up the same amount of medals.
So taking this all to account it looks highly unlikely that the rest of the UK isn't really going to be effected by Scotland becoming an Independent Country.
*For this very unscientific post I have awarded all the gold medals won by a Scottish individual to Scotland and made the assumption that the teams with a Scottish member in would not have achieved gold if they were replaced with an athlete from one of the UK's three other countries.