I found the same John Plimmer you are speaking of in 1870 in the US in LaSalle IL.
There is a family in England that might be related.
Name Age in 1871 Birthplace Relationship Civil Parish County/Island .
Enoch Plimmer 27 Wrockwardine, Shropshire, England Son Wrockwardine Shropshire
John Plimmer 66 Wrockwardine, Shropshire, England Head Wrockwardine Shropshire
John Plimmer 16 Wrockwardine, Shropshire, England Son Wrockwardine Shropshire
Sarah A Plimmer 26 Wrockwardine, Shropshire, England Daughter-in-law Wrockwardine Shropshire
Civil parish: Wrockwardine
Ecclesiastical parish: Wrockwardine Wood
County/Island: Shropshire
Country: England
Source information: RG10/2809
Registration district: Wellington
Sub-registration district: Wombridge
ED, institution, or vessel: 2
Folio: 51
Page: 35
Household schedule number: 168
In 1881
Name Estimated Birth Year Birthplace Relationship Residence .
Enoch Plimmer abt 1844 Wrockwardine, Shropshire, England Son Wrockwardine
John Plimmer abt 1805 Wrockwardine, Shropshire, England Head Wrockwardine
Sarah Plimmer abt 1843 Lilleshall, Shropshire, England Daughter in Law Wrockwardine
Margaret Thomas abt 1870 Lilleshall, Shropshire, England Visitor Wrockwardine
Address: Nabb
Civil parish: Wrockwardine
County/Island: Shropshire
Source information: RG11/2681
Registration district: Wellington
Sub-registration district: Wombridge
ED, institution, or vessel: 2
Folio: 56
Page: 33
In the 1900 US census, there is info on the IL connection. Immigration Etc. I have attached it below.