The earliest British bottleneck cartridge I could find in my COTW,3d Ed. is the .500/.450 No. 2 Musket, from "the early 1870's"
In the same publication, the .56-46 Spencer is shown as being "introduced" in 1868, but it must have been developed earlier. I can't see when bottleneck sporting cartridges were introduced in Europe, but several Sauer cartridges look quite old but no dates are shown.
The earliest centerfire bottleneck round I could find in COTW is the 10.75 x 58R Russian Berdan adopted in 1868. At least it seems the idea is very likely American