Ms Nielsen had been working as a missionary at the Alpine German-Speaking Mission since July 2019.
Church officials issued a statement on Tuesday night addressing the death.
“We are deeply saddened to share news of the passing of a young missionary serving in Switzerland,” Daniel Woodruff, a spokesman for the LDS Church, said.
The church also offered its condolences to the family.
“We pray they will feel the peace and comfort of our loving Heavenly Father as they deal with this tragedy and honour her life,” Mr Woodruff said.
Ms Nielsen’s family told the Deseret News that they were struggling but trying to remain faithful.
“Annabelle was a sweet, sweet young girl, just a wonderful young woman,” Rodger Lyman, a spokesman for the family, said. “We’ve known her almost her whole life. Just fun, fun to be around, very kind and loving, very active.
“The family is struggling, as anyone would, but keeping the faith.”
As more people turn to outdoor activities to provide recreation amid the coronavirus pandemic, some accidents have occurred.
Last month, a Washington state-based pilot in the US Navy died during a hiking accident.
Lt. Adam Johnson, 29, disappeared on 26 Sept. while hiking near Vesper Peak in the Mount Baker-Snoqualmie National Forest.
Search and rescue teams found Mr Johnson’s body in a swift-moving creek the day after he was expected to return home.
The Snohomish County Medical Examiner’s autopsy ruled that Mr Johnson died from drowning and multiple blunt force injuries that were ruled accidental.