Disclaimer: this is just mere speculation, but is heavily supported through cannoical references.
Aside from knowing Karin and Nagato are descendants of the Uzamaki clan, the author probably chose to leave out "Uzamaki" as a surname for them because it is not expicilty stated their surname is Uzamaki. It could have been something entirely different.
If you follow the history of the Uzamaki clan, they were scattered after the destruction of their homeland Uzushiogakure. Uzushiogakure's destruction happened roughly around the time Kushina moved to Konoha.
Karin is only ~17 years old. Kushina moved to Konoha roughly at the age of 12, and died at the age of 24. Given that Karin is roughly the same age as Naruto, this means she was not even born yet when Uzushiogakure was destroyed. We know very little of Karin's life at a young age and her parents, but it is stated Karin grew up as a member of Kusagakure (Village Hidden in the Grass). This means at least one of her parents moved to Kusagakure, and possibly married in another clan.
Nagato is stated to be 35 at the time of his death, meaning he was alive prior or during Uzushiogakure's destruction. However, he grew up in Amegakure (Village Hidden in the Rain). We are shown small glimpses of Nagato's parents in the manga and anime, however we only know at least one of his parents was an Uzamaki. Just like in Karin's case, one of his parents could have moved away from Uzushiogakure and possibly married into another clan.
The same can also be true for both Tsunade and Gaara, however we are not shown enough history for them to make an accurate assumption