How do SIGMETs arrive in a cockpit on a modern longhaul flight?
This varies between airlines. In my company (large European Carrier) SIGMETs are uplinked to us via ACARS and then automatically printed on the cockpit printer. 

The differences in regions is the format of the SIGMET. In the USA the points of a polygon are mostly described as list of DIR/DIST of VORs (eg 15NM NE STL). Very hard to read if you are unfamiliar with the area. Rest of the world uses coordinates (N3905W09046). Way better to locate on a chart. 

There are efforts to automatically download (via satellite) and parse SIGMETs and display them in our electronic charting application in the aircraft EFB. This works currently so-so as there are surprisingly often typos or similar mistakes in a SIGMET text which kills the parser.