I know that living on an island and not being invaded by Napoleon greatly contributed to this, but why does every single other country in Europe has a system of civil law, instead of some other third system, or some closer to the UK*'s (with a stronger emphasis on case law)?
*Including Scotland only in part, since most sources says they have a "hybrid or mixed legal system" (involving both common and civil law).
Well, you hit it on the nose, Napoleon didn't invade The British Isles. Many call Napoleon a conquerer or a tyrant but in reality, he exposed Continental Europe to the Code Napoleon which in turn was adopted in varying degrees to various nations.
While not all of Europe had been under the Napoleonic code, many did adopt it due to alliance or direct rule, such as what is now a majority of Germany, Belgium, Spain, Italy, Croatia, Poland, and the Netherlands. However I cannot go beyond that since I'm not well versed in the legal history of the mentioned countries.