I found him in Sweden.
According to his 1890 marriage certificate from Quincy, Norfolk, Massachusetts, his parents were named John (likely Johan, if he was Swedish) Nelson and Johanna.
On Ancestry I found a scan of his naturalization petition for Frank W. Nelson from 26 Nov 1891 in Quincy, MA. In the petition, he states that he is a polisher by trade, born in Sweden on 28 Dec 1859, and that he arrived in the USA via Boston on 26 Jun 1887. FamilySearch shows an index card with similar info.
I plugged his birthdate into ArkivDigital and found a likely match. There's a Frans Vilhelm with the same birthdate in Göteborg. (Here's his birth record in Göteborg's Hospitalförsamling.) His parents are Johan Alfrid Nilsson (b. 6 Oct 1824) and Johanna Andreasdotter (b. 23 Mar 1825 in Wänga).
This Frans Vilhelm Nilsson appears in Göteborg's Gamlestad in the 1888-1889 churchbook. It seems he married Lovisa Johansdotter on 5 May 1885. They have a son Carl Johan (b. 27 Jul 1886). Lovisa dies that same day. Frans Vilhelm leaves his parish for North America on 31 May 1887. His son Carl Johan becomes the foster-son of Frans Vilhelm's older brother Johan Teodor (b. 18 Apr 1854). Johan Teodor leaves for North America on 27 Jun 1888,. His wife Maja Alfrida Jansdotter (b. 8 Nov 1858) follows with their kids (along with Carl Johan) on 2 Jul 1889.
Indeed, in the 1900 US Census (which Wayne referred to in his answer) you can find Frank W. Nelson in Quincy, MA with his second wife Annette, their kids Ester and Ernest, and Carl J Nelson (born May 1886, Sweden). The entry below them is Frank's brother John T. Nelson and family.
It's super nice and clean when the documents and records all support and agree with each other.