I suspect that the groupings will fall in order of:
First world nations doing it themselves. In the current world this is the United States and China
Economic, military or other formal alliances like NATO, the EU or the The Shanghai Cooperation Organization.
"Second" world powers like the BRIC nations (less China, now)
Informal groupings like the "Aglosphere".
The reasons for these groupings are simply put, cultural. IF you are heading out on a colonization voyage, you want to settle with other people of the same cultural background. Think of the great age of exploration. Spanish and Portuguese explorers and colonizers were great rivals, despite the similarities between the tow nations. When the English began colonizing the New World, different sub groups of Englishmen would actually settle in different areas of North America, despite the potential need to band together against common enemies like the Natives, the French or the Spanish. Hence the first settlements of Puritans was based on religion, while following groups included Protestant dissenters, who created the political and social institutions that underpin the modern United States. You will note that other groups of English settlers with different cultural backgrounds created different political institutions in Canada and Australia.
The other reason, which is related, is a starship would be an enormous engine of destruction in the wrong hands. Current technology would support an ORION nuclear pulse drive, and it is unlikely that anyone will want to share the resources when they include thousands of nuclear "pulse units" AKA bombs. The military resources of the sponsoring nations will be going to producing, protecting and monitoring the pulse units of their own ships, and watchfully observing anyone else to ensure their "pulse units" are safely stowed aboard the ships and not potential threats to the homeland.
It seems also certain that the challenge of building a starship will actually be too much for many of these groupings. Once again, part of it is the resource base, but the other part can be described as institutional culture as well. The United States has a very competitive high tech sector, so companies as varied as Boeing, SpaceX and Blue Origin can all be competing to create technologies and provide services for the starship project, SpaceX charges @ 60 million USD for a launch on the Falcon 9, while a similar launch on a ULA Delta is almost 400 million USD, so you see how competition works in your favour. The ESA, OTOH, takes forever to come up with new generations of rockets, and building one is a maze of securing contracts for every member nation. Russia, China and India can be considered intermediate cases.
Because of both the natural tendencies for people to align within natural, cultural or nationalist groupings, as well as the potential threat of a starship in orbit, you may be more likely to see espionage, sabotage and unbridled competition between the nations as they scramble to complete their ships and exit the Solar System.