Find the estimated population of each county in Nebraska at the American Community Survey - Data Profiles. Select the year you want. Under "Select a Geography Type", select "County". A box will pop up for you to select the state. Then another box will pop up for you to select the county. Then click on "Get Data Profile Links". A listing will pop up. Choose "Social Characteristics". Nebraska and your chosen county will open. Scroll way down to find "Place of Birth". "Total population" as of 2021 is the next line.
Additional information is also in the historical population counts (1900 to 1990) for all counties in Nebraska text file.