Does Sheffield only process the United States ?
No, they process applications from countries all over the world. If they only processed applications from the US, everyone would probably get their visas in 1-2 weeks.
They started processing US settlement visa applications in March 2013, along with applications from Canada and a couple of other countries.
However, they have been adding more countries since then and between January and March 2017, they moved several other countries to Sheffield (at least 20 countries).
The problem was that Sheffield did not have suitable resources to deal with them and they became overwhelmed. As a result, the processing times went from 2-5 days priority and 1-2 weeks non-priority in February, to 2-3 months priority and 3-6 months non-priority this summer, and they're only just clearing the backlog now.
Between July and September 2017, UKVI processed 13,067 settlement visas in total around the world (and granted 10,254 of them). I would estimate that the majority of those were processed in Sheffield. Only 773 of those 10,254 approved applications were from US citizens (I'm not sure how many total settlement applications were made by US citizens though, but the approval rate is around 95-98%).