The real question would be not "how to implement common law," but "how to prevent codified law."
For the better part of post-Roman Empire European history, the only codified source of law was Roman Law, and that was in most places and times only used as something to fall back on if local customary law did not apply to the case at hand.
Only with the Enlightenment did it become a very popular idea to codify law along 'natural' and 'rational' lines, which did away with customary law for the most part. The French really took to it, Napoleon implemented his civil code everywhere he went (which was basically everywhere in Europe), and post-war most re-instated rulers found out such a centralized, government-controlled legal system was very useful to them. So it stayed.
The trick is to prevent this development. Keep in mind that even before the French revolution, various Enlightened despots were already implementing codes of law, although they were often far from universal. (Examples include Frederick the Great of Prussia, Catherine the Great of Russia, and Maria Theresa of Austria.) This might mean that the trend gets started anyway, wether you have a French revolution or not. On the other hand, Britain stuck to common law partially because they wanted to be not like France.
Since preventing the Enlightenment is very difficult and entails huge swarms of butterflies the size of pterodactyls, I'd recommend having the French Revolution take place, and then quickly go horribly wrong. Pretty much OTL, but then they get crushed at the very height of the terror. Result: that's what everyone remembers. Other European nations will want to be 'not like France' as well, just like Britain IOTL. They'll be very hesitant to embrace 'French' legal ideas.
Extra reason for that: Montesquieu's ideas on the codification of law were actually quite impractical, because he died before they were ever implemented. When the French revolutionaries implemented them just as he had proposed, this led to enormous administrative hassle. The problem was that Montesquieu had written that judges should only read the law, and should never be allowed to interpret it. Since nearly every existing word can be interpreted in multiple ways, reaching a verdict became next to impossible, as judges were terrified of being accused of 'illegally interpreting the law'. If that's what people in Europe generally associate with codified law, they'll not be eager to adopt it.
Thus, customary law stays. Like common law, it relies heavily on precedent and tradition. With the advent of modern communication, it will become less 'local,' and various local customs will also start to influence each other. I expect the result to be something very much like common law.