The Law for the Protection of German Blood and German Honor states:
Marriages between Jews and citizens of German or related blood are forbidden.
Sexual relations between Jews and citizens of German blood are forbidden.
Assume that a Jewish husband (does not practice the Jewish religion but is classified as a Jew by parents/grandparents) and a non-Jewish wife (Protestant) are married and have kids, before the Nuremberg Laws have come out.
Did these new Nuremberg Laws preventing marriages/sexual relations between Jews and "citizens of German or related blood" (wife) split up existing marriages between Jews and non-Jews?
It is very rare for new laws to apply retroactively, even Hitler's Nuremberg Laws. In this case, the prohibition of mixed marriages did not apply to couples who were already married. However, as the paper linked above observes, the Law for the Protection of German Blood and German Honour did include one provision in regard to existing marriages:
"Proceedings for annulment may be initiated only by the Public Prosecutor."
So, while in general the Nuremberg Laws did not split up existing marriages between Jews and non-Jews, they did include the provision enabling the Public Prosecutor to do so.
Yes, they did, in an "indirect" way. This was particularly true after a 1938 supplement made the Jewish partner's racial status grounds for divorce by itself.
That is, the laws were designed to allow, and even encourage "mixed" couples to break up. In a book of the same title, a "Mischling, Second Degree,", Ilse Koehn relates how her German mother was induced to divorce her (half) Jewish father in this way.
Not even the Nazis wanted to "force" the breakup of these marriages ex post facto. That would have been too socially disruptive. In fact, during the Rosenstrasse protest, German wives fought and won a battle to save their Jewish husbands from arrest. It was "enough" for the Nazis to prohibit such marriages going forward from the Nuremberg laws.
If you're asking about the direct legal consequences of the "Blutschutzgesetz" (Law for the Protection of German Blood and German Honor) - it did not apply retroactively, no.
The declaration that marriages entered into being null only applied to marriages entered into despite the law, i. e. after it took effect. (And technically, the last sentence about the public procecutor meant that the spouses could not file action for annulment themselves).
For reference, here is my rough translation of § 1 of the Blutschutzgesetz:
(1) Weddings between Jews and citizens of German or similar blood are prohibited. Marriages entered into in spite [of the prohibition] are null, even if they have been entered into in a foreign country to circumvent this law. (2) Proceedings for annulment may be initiated only by the Public Prosecutor.
While the legal status of existing marriages was not changed, the laws introduced the notion that the government had an urgent interest in protecting "German blood" and that intercourse between "jews" and "Germans" was a danger to important public interests.
We shouldn't underestimate the damaging social and psychological effects of these laws.
It is important, too, not to overestimate the exact words of NS laws. The intent was well understood by the bureaucracy and the courts.
Also relevant: The other "Nuremberg law", Reichsbürgergesetz, which did affect mixed-"race" families, and the 1938 "Ehegesetz" that declared the racial status of "Jewish" spouse reason for divorce.
I wanted to post this as a comment, but unfortunately cannot due to low rep (new acc).
The accepted answer is perfectly correct. This situation applied to my great-grandparents (husband Arian, wife Jewish). He was heavily pressurised into divorce, but refused to do so. Both being artists, they suffered heavily from being banned from their profession. At least they survived, including all of their six children, one of them my grandfather.
They recovered after the war and my great-grandmother lived to a very old age (1901 - 98). I got to know her as a child, a wonderful woman. Ironically my great-grandfather was killed by a drunk driving GI in the 70s. I wish I had got the chance to meet him :(
The Word used to describe the prohibited act is "Eheschliessungen", which explicitly refers to the act of wedding, not to a marriage in itself. Had the word been "Ehen", it would have been ambigous.