It's not quite correct to say that Scots law is entirely independent though. It's true that The High Court of Justiciary is the highest court of appeal for Scottish criminal cases, but the Supreme Court of the UK (formerly the House of Lords) is the highest court of appeal for civil cases in the UK and decisions in Scotland can be appealed there. However, the rules and procedures of Scots law are different and so judgements in England which are used as precedents in Scotland have to be reinterpreted in the light of Scots law and visa versa. I know for a fact that there is a standard contract clause which
currently has a different interpretation in England versus Scotland because neither of the interpretations have yet been appealed in the Supreme Court. As you can imagine this can cause endless amounts of expensive legal haggling. Scots and English law also use very different legal terminology and procedures for many everyday processes. Interestingly, this is because Scots law is a mixture of civil law derived from Roman law (like most of Europe) and common law (like England, US, Australia, Canada etc).
Also, I think English law can in theory draw on precedents set by courts in other commonwealth countries; I vaguely remember seeing Australian case-law cited when I was studying law related to professional negligence, but I can't find a reference to it now.
It's a very typically British setup which is the culmination of 100's of years of compromises here and there which viewed together now are totally impenetrable to the layman. The whole setup is utterly confusing and should be thrown away and rewritten from scratch preferably with the same law applying over the whole of the EU if you ask me, but then I'm not a lawyer.