Legendary gunfighter Ben Thompson comes to mind. At the age of seven, he immigrated to America from his birthplace of Knottingley, West Yorkshire, in England. After numerous scrapes with the law, he became city marshal of Austin, Texas, in 1881. Despite accounts reporting he was mainly drunk on duty, he was re-elected.
The Meagher twins, Mike and John, were born in Ireland in 1843 and moved to the U.S. after the Irish potato famines in 1845-52. In 1871, Mike became marshal in Wichita, Kansas, where he gained a reputation as an outstanding lawman. His brother not only worked for him as a deputy, but so did Wyatt Earp.
These are just a couple, among probably many others, from the United Kingdom who put on a badge.
If you have a question, write: Ask the Marshall, P.O. Box 8008, Cave Creek, AZ 85327 or e-mail him at firstname.lastname@example.org