If we believe Harlan Ritchie, and why wouldn't we as he is one of the beef gurus, the existence of purebreds depend on your definition of purebred
Shorthorn breed created with the best native Shorthorns in NW England n with a titch of Galloway
Hereford breed created by infusing the native cattle of Herefordshire in SW England with Flemish cattle
Angus (or is it the Angus) had a titch of Shorthorn in the beginning
Red Poll had a titch of Galloway and Devon breeding
As near as can be determined only the Galloway and Highland of the British breeds have remained free of outside blood
Of the Continental breeds the US Charolais didn't come directly from France but via Mexico in the 1930s - these cattle as well as those in Canada from France were crossed with native cattle. Charolais in the US today primarily the result of upgrading with Angus, Hereford and Shorthorn
In the 60s and early 70s a wave of bulls from other Continental breeds were brought into Canada and used in the upgrading process on British cows - thsi includes Simmie, Limmie, Maines, Salars, Chianina, Piedmontese, Belgian Blue, Romagnola, Marchigiana, Tarentaise, & Normande. After 4 to 5 generations of upgrading these new breed associations registered these animals as purebreds - they still had a small amount of genetic material from the British base cows.
Other tidbits include the chaos of Europe after WWII leading to cattle running loose and a significant amount of inbreeding and the unethical (his word) infusion of Holstein and Brown Swiss into the Angus and Simmie into the Hereford to follow the fad changing short fat cattle to the more leaner taller type.
And finally in the 70s the ASA established an appendix program in which it opened the herd book to Milking Shorthorns, Irish Shorthorns, Australian Illawara Shorthorn and Maine-Anjou
(Info from an article by Harlan Ritchie and Roy Wallace "What is a breed?" Beef July 2006 - if you want a copy I can send a pdf)
so basically we almost all have mostly mutt cows - some with papers - is this like a Labordoodle or a Peekapoo??