Around 1910, the park fell into disuse and became a wild swamp. Around this time, Mary Carter, Nielsen's daughter, moved the swing located at the Grove up to her own property in Provo Canyon at Vivian Park. Here is a picture of the swing in 1916 in Vivian Park.
Used by permission, Utah State Historical Society, all rights reserved.
Nielsen originally constructed the swing, intending it to be a combination of a swing and a carousel. Twelve swings hung down from the top. Located at the center was a cross beam which allowed either a couple of people or a horse to turn the swing in a circle. This allowed the swings to lift to the side in addition to back and forth.
I could not find any information about what happened to the original swing. However, when the city of Orem purchased Nielsen's Grove in 1995, they reconstructed the swing at the park in Orem. Below is the current state of the swing.
There are only three swings currently rather than 12 but every other aspect is quite similar to the original. If you have never seen this park, you really should go check it out. Orem has done a great job recreating Nielsen's original ideas, so much that the Daily Herald stated that the park is "a little-known gem in Orem."
If you want more information about Nielsen's Grove, the city of Orem has created a museum dedicated to the life of Nielsen. The museum is located in a replica of Nielsen's old farmhouse and is located at the park. At the museum, you can find replicas of some of the statues that were located in the Grove, an old painting of the park, and lots of additional information regarding the life of Jorgen Nielsen.