The law you are referencing is the Law of Return, and the definition of using 1 Jewish grandparent to establish the right to immigrate was passed in 1970, after the founding of the state. This was in fact adopted from the Nuremberg definition, on purpose.

Firstly I want to point out that the terminology here in the original question is incorrect, although this is somewhat confusing to people not acquainted with it. The Law of Return is not who is considered Jewish by the Rabbinate, which is still done according to a traditional (Orthodox) understanding of Jewishness. The Law of Return is not who is Jewish then, but who is allowed to immigrate to Israel. Although converts from non-Orthodox movements are also allowed to immigrate to Israel which can further confuse those not acquainted with the nuances.

Non-Jewish immigration is covered under the Nationality Law, made in 1952.

The initial statement in the Declaration declared that Israel was a place “for Jewish immigration and for the ingathering of the exiles” although it wasn't until 1950 that the law was created on it. On July 5, 1950, the law was passed that allowed for any Jew to have "the right to come to this country as an oleh”, Oleh being the Hebrew word for immigrant.

The 1950 law made no mention of who was considered Jewish for the purposes of immigration. In 1970 the law was amended to allow anyone with one Jewish grandparent, or married to a Jew to immigrate. This expansion of the law was indeed based on the Nuremberg definition of Jewishness. The State of Israel wanted to be a haven for any Jew who could face persecution.

Here is the text of the amendment (amendment 2, 4a):

"The rights of a Jew under this Law and the rights of an oleh under the Nationality Law, 5712-1952, as well as the rights of an oleh under any other enactment, are also vested in a child and a grandchild of a Jew, the spouse of a Jew, the spouse of a child of a Jew and the spouse of a grandchild of a Jew, except for a person who has been a Jew and has voluntarily changed his/her religion."

except for a person who has been a Jew and has voluntarily changed his/her religion."*

As a side note here, this was actually a deviation from the Nuremberg laws as the Nazis still considered those who converted/were practicing another religion to be Jewish. They would search Church records to find converts to Christianity and also persecute them.

Another factor that plays in here is the fate of the Jews in the USSR, there were many Jews attempting to immigrate to Israel who were denied the ability to do so by the Soviets. These people were called Refuseniks, the name derived from the refusal they received.

The Soviets since the time they inherited the Pale of Settlement, and the Jewish populations therein had their own 'Jewish Question' and made many attempts to both secularize and eradicate Jewish culture. Initially, the Tsarist regulations were dropped, and Jews were given emancipated in the new Soviet state, the Bolsiveks both eradicated the discriminatory laws on jews but also made several attempts to remove Judaism and Jewish culture.

Conditions worsened under both Stalin and Brezhnev, and many Jews wanted to immigrate to Israel. However, since the Russians had suppressed their culture many were not, or could not show Jewishness under the traditional definition. However, the recasting of the definition to that of the Nuremberg Laws allowed many of these refuseniks to immigrate although it would take time for them to get complete permission from the Soviets to do so, primarily under Gorbachev when he agreed to the protocols of the OCSE (Organization for Security and Co-operation in Europe).

The estimated Jewish population in 1970 in the USSR was 2.1M reduced to around a quarter of a million by 2019. The immigrants went to three countries primarily, Germany, the US, and Israel with the bulk going to Israel. Many Jews from Poland (then under Soviet control), after many antisemitic pogroms were carried out in 1968 maybe have also influenced the writing of the law as they were looking to emigrate to Israel.

There is a (PDF) chart here on page 2 from a presentation at Harvard on the Russian Jewish immigration.

The control of who is 'officially' Jewish by the Rabbinate is the result of a compromise prior to the founding of Israel in that the British adopted the Ottoman system with the Palestine Order in Council, 1922.

Sources:

Jewish Emancipation by David Sorkin The Jews of the Soviet Union by Pinkus