This largely depends on your political leaning more than it depends upon fact.
We can in fact use the European Union as a sort of realistic example of the benefits and downsides of such a system.
Furthermore, I would like to stress. The most essential omission from the question is that of the level of autonomy that each of the existing countries, groups and factions would have under such a federal government. As such, I’m assuming that we will have a high level of autonomy. And that we are merely talking about the economics and politics of such a situation.
If you wanted to take this combination to medium autonomy, you might base this off of the structure of Imperial Britain. Or if you longed for there to be low autonomy and an extremely centralised government then you could base this off the US federal system.
But, in the meantime, we shall keep it at high autonomy.
Our world is already more connected than it ever has actually remained in all of human history. This would speed up the pattern of interconnectedness and cause a fantastic increase in wealth throughout the world, specifically for poor countries. Poor countries would have increased access to the wealth of the developed world. While this would bring substantial investment opportunities for the middle and upper classes in the developed world; it would practically totally destroy the working class in these nations.
We have seen this happen prior to. Particularly in Britain, where increasing wages and a more competitive developing world destroyed the standard market in the rural mining communities in Wales and the North of England.
As an outcome, anticipate short, but most likely, medium-term political instability paired with an increase of anti-globalist sentiment.
When again, this can be seen in a lesser kind today. Where after the crisis of 2008, numerous working people were struck hard and the middle class was badly damaged. The consequence of this is that the wealthiest in society handled to get richer, while the poor saw their opportunities shrink. Lots of people would get frustrated at migrants coming and working in their own nations.
Matters of immigration are by nature, highly questionable. While many argue that immigration is much better for society in terms of economics, and indeed this is true, the primary issue comes at the cultural level.
Greater than sustainable levels of migration from greatly different cultures frequently cause fragmentation within the more comprehensive society. Various cultures who do not always get along will be pushed into close communities within cities. This is taking place today. And will likely far exceed assimilation level causing cultural pockets to express themselves in various levels of society.
This will likely be the biggest issue facing a global government.
The problem being, that people wish to cope with those they appear as being from the exact same tribe.
For a global federal government to prosper, there would have to be a body that regulates the cultural makeup of different areas of the polity lest civil motions tear the union up from the inside.
This is a difficult one.
Basically, I personally would regard an international currency as a disaster.
It would in impact, be a form of international feudalism.
It would avoid bad nations from cheapening their currency to attract financial investment, keeping them poor.
I’m not precisely sure on the utility of this, as this has been recommended for the European Union – however it is possible to have a worldwide currency with various levels.
I.e. a worldwide currency for rich nations, one for middle-income nations and so on
Regardless, the concern then turns to trade routes and human capital.
Do we keep things the same? Where would the centres of administration be?
London, New York City, and others are excellent contenders. But then undoubtedly a country like China would feel overlooked.
This is the challenging part. When economics satisfies politics.
This is completely hypothetical.
And then, while the world does appear like it is going towards a more globalised and integrated society, how does this happened?
A fairly loose union of all of the nations is possible. Anything similar to the worldwide governments from sci-fi books and movies is impossible to bring about without the usage of force.