The older answers already mention one of the most important parts of the answer:

The law in positive law countries (and in others, too) consists of many hundreds, or even thousands different codes, of which most are not at all or only marginally changed, even if a country dissolves completely or switches its form of government, eg from absolute monarchy to democracy.

I suppose, that you are interested in the continuity of the law and the lawyers in the case of countries dissolving or changing the form of government, and you use Russia and Yugoslavia as examples of countries with such events in their history. I am going to use mostly Austria-Hungary/(German) Austria/Nazi-Germany (Deutsches Reich)/Austria as an example because that is what I know best.

Using the older answers and what I am writing further below, some general statements about the fluidity of the law are arguable: Laws get changed, sometimes a lot, and not only during big changes like revolutions. As a whole, the law is nevertheless rather static, even during big changes.

A lawyer, be they attorney, judge or public servant, will have to familiarize themselves with new laws, codes and court decisions repeatedly throughout their professional life all the time, while basic principles of the legal system generally remain the same. They should have learned how to do that at university, they can use the legal text themselves, they can use explanatory remarks by the legislator (if they exist) and articles, books and legal commentaries are quickly available, sometimes even before the law in question enters into force.

In a revolution, transition etc the first law to be changed is usually the constitution. That is the common name for the type of law which regulates, how (and by whom) laws are made - an absolute monarch, the people, parliament and so on. Many everyday rules, like who has the right of way on a crossroad or if a contract for buying a house has to be in writing does not conflict with whether the Tsar or the people decide, and those laws are kept.

Different codes of law exist. Some deal with the whole or most of the civil law, contracts, marriage, inheritance, tort law, rent, labour law, civil status and so on. Well known examples are the French Code Civil or the German Bürgerliches Gesetzbuch (entered into force 1900 and is still applicable). I'll use the Austrian civil code, the ABGB (Allgemeines Bürgerliches Gesetzbuch) as a more specific example for continuity over centuries, but first another step back:

The basics of civil law, especially of contract and tort law, stem from Roman Law. And those basics have been part of most continental legal systems since then. Eg for a purchase contract the parties have to agree upon the Essentialia negotii (the essential parts of the contract), which is what is bought and what's the price. Many details change over time, eg equal rights for women, certain aspects of the sale of consumer goods and associated guarantees (Directive 99/44/EC of the European Parliament and of the Council of 25 May 1999).

Back to the Austrian civil code, ABGB: It came into power in 1811 and dealt with almost all parts of Civil law. Austria was called Kaisterthum Österreich (Austrian Empire) then (not even Austria-Hungary yet), and was an absolute monarchy, Franz II/I. who recently had dissolved the Holy Roman Empire was Austrian emperor.

Today, the ABGB is still applicable law in Austria and was it during times of absolute und constitutional monarchy, democracy and fascism. It was also applicable in many other countries for some time, when Austria-Hungary dissolved after 1918. (1938 to 1945 Austria was part of Nazi-Germany and had the BGB. One time the ABGB was accidentally rescinded). Drastic reforms were made during WWI and in the 1970s.

So a lawyer in the Austrian Empire, Austria-Hungary, (German) Austria, (fascist) Bundesstaat Österreich, after WWII and after Austria joined the EU till today could continue in their job as an attorney, judge etc through all these changes, working with the same legal text.

Some fields of the law might develop completely new, like divorce law or more recently environmental law, but not necessarily in conjunction with sudden, big changes. But as in many other professions, lawyers must keep up with the changes in their work environment.

However, individual lawyers can be severely affected by such changes, for example, most members of the constitutional court of Austria were ousted when Austria became a fascist state (so they could not strike down the unconstitutional laws), and after 1938 many lawyers were sent to the concentration camps like Heinrich Klang, who survived Terezin and became a judge at the Civil Supreme Court (OGH) after WWII. Others could leave Europe, like Hans Kelsen, "architect" of the Austrian constitution (wich is also still in force today), who ended up in Berkely.

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Literature:

Barbara DÖLEMEYER, Die Revision des ABGB durch die drei Teilnovellen von 1914, 1915 und 1916, in: Ius commune VI, Frankfurt am Main 1977, 274-303;

KAMILA STAUDIGL-CIECHOWICZ, Österreich: Das österreichische Zivilrecht und die österreichische Zivilrechtswissenschaft in der Zwischenkriegszeit, in: MARTIN LÖHNIG / STEPHAN WAGNER (Hrsg.),„Nichtgeborene Kinder des Liberalismus“? Zivilgesetzgebung im Mitteleuropa der Zwischenkriegszeit(Tübingen 2018), 39–69.

KAMILA STAUDIGL-CIECHOWICZ, Die „Säuberung“ der Universitäten 1938 am Beispiel der Wiener Rechtswissenschaftlichen Fakultät In: Journal on European History of Law 9 (2018), Nr. 2, 40–49.

Thomas Olechowski et al Die Wiener Rechts- und Staatswissenschaftliche Fakultät 1918–1938 Schriften des Archivs der Universität Wien, Bd. 20, 2014 - https://www.vandenhoeck-ruprecht-verlage.com/downloads/openAccess/OA_978-3-89971-985-7.pdf

Ilse Reiter-Zatloukal, Barbara Sauer: Advokaten 1938. Das Schicksal der in den Jahren 1938 bis 1945 verfolgten österreichischen Rechtsanwältinnen und Rechtsanwälte, hrsg. vom Verein zur Er-forschung der anwaltlichen Berufsgeschichte der zwischen 1938 und 1945 diskreditierten Mitglie-der der österreichischen Rechtsanwaltskammern, Wien: Manz 2010

Thomas Olechowski, Hans Kelsen - Biographie eines Rechtswissenschaftlers, Mohr-Siebeck, Tübingen 2020

Thomas Olechowski Rechtsgeschichte Einführung in die historischen Grundlagen des Rechts, 5. Auflage, Wien 2019

Introduction to legal science, as taught at the University of Vienna: https://ssc-rechtswissenschaften.univie.ac.at/diplomstudium-doktoratsstudium-und-ec/studium/studienanfaengerinnen/steop/einfuehrung-in-die-rechtswissenschaften/

Austrian examples of transitional laws: https://www.ris.bka.gv.at/GeltendeFassung.wxe?Abfrage=Bundesnormen&Gesetzesnummer=10000201 https://www.ris.bka.gv.at/GeltendeFassung.wxe?Abfrage=Bundesnormen&Gesetzesnummer=10000078

edit: Thanks, kind kind stranger, for my first ever award. I also took the opportunity to correct a few typos and linguistic errors.

While waiting for an answer to this question, feel free to have a look at /u/Kochevnik81's answer to the question What did lawyers and Judges in the USSR do when the Soviet Union fell and the laws were all changed?