I'm afraid the timeline is a bit off here...Justinian's code never really disappeared in Europe, and there were other legal codes that still existed as well (notably the Theodosian Code). Naturally Justinian's law code was still in use in the eastern Roman Empire, although it was adapted by later emperors into new law codes. In the former western part of the Empire, there were now Germanic law codes, but they were influenced by Roman law too, as was the canon law of the church. They were used alongside Roman law. In places where the Empire had been the strongest and most urbanized (Italy, southern France for example), Justinian's code and other collections of Roman law were always in use.

Sometimes we talk about a "rediscovery" of Roman law in the 11th century, which is sort of accurate, since that's when a complete text of the laws was more or less available to everyone again. It had been in Bologna the whole time and Bologna became the centre of legal scholarship in western Europe. Certainly by the 12th century it was well-known, long before the Renaissance (unless of course you mean the greatest of all renaissances, the Renaissance of the Twelfth Century!)

I'm sure we could make the argument that Roman law had an enormous influence on early modern legal scholars...someone who works on more recent history might be able to help out with that. But I wrote something about the use of Roman law in the Middle Ages that might help with the medieval background:

How did Roman Law start to be seen as obligatory in Mediaeval Europe?

Firstly I agree that Roman law hardly was totally lost or gone in Western Europe (it certainly wasn’t in the Eastern Roman Empire which existed throughout most of the medieval age). This can also be seen in canonical law which still bore a healthy influence of Roman law tradition.

That being said the questions seems to be how the infusion affected European legal systems. In my opinion the influence can hardly be overestimated. Especially (but not only) the continental legal systems (Code Civil tradition, BGB/ABGB/ZGB Tradition) are based on the tradition of reception of Roman law (though of course modified by Germanic and some other influences).

The question you asked has been the topic of a virtual slew of works by legal historians and I will in order to not get too lengthy make some statements in regards to German legal development since I’m also most familiar with these.

After the complete copy of the Corpus Iuris Civilis (Justinians Code) was rediscovered in the West and used as a foundation of the influential Bologna law school it kicked of academic interest and university based education for legal practitioners across Europe.

The first major step were the Glossatores (Irnerius, Azo, Accursius, Baldus di Ubaldis) who commented on the digests by writing in between or around the Digesten. An example from Azos work can be found here:

https://en.m.wikipedia.org/wiki/Azo_of_Bologna#/media/File%3ADecretals_glossa.jpg

By the way: a proverb from that time was „si non hada Azo - non vada in palazzo“ (if you don’t have the commentary of Azo don’t bother to go to the court house).

From the glossators onwards the Usus Modernus Pandectarum developed with important figures like Bartolus and later among others Corning and Carpzov. The Usus modernus was a reception of the Roman law and the glossators. It incorporated later legislation and also in parts deviated from Roman law as it was practiced before. Yet the heart and soul of civil law and constitutional law (and to a lesser degree even criminal law) remained the CIC and the works of the classic legal thinking.

The question whether this thinking really was „Roman“ thinking is hotly debated among legal scholars (romanists) to this day. Some think it was, others believe it was „watered down“, still others take the position - which I tend to endorse - that it was a continuous development and evolution of legal thinking which adapted to society and historic circumstances.

Please note that the idea of „the“ classical legal thought is also a construct. The legal thinking of Rome developed over centuries and though certain foundations and basics seem to be widespread there are quite a few differences along the centuries.

Now the Usus Modernus was effectively used from the 16th to 18th century and it was replaced in Germany by the historic school which actually re-focused on original Roman law and the CIC. From these the codification movement began with early ones in Austria and esp France and later Germany and the BGB which, as some scholars say, is up to 90% Roman law (as adapted and developed of course).

Therefore: the legal thinking of the continental tradition has been mostly shaped by the influence of Roman law.