It won't work very well, just like how the UN isn't actually capable of doing much other than maybe writing very angry letters.
The first problem is funding. If a country contributes more to SWORD they expect to have greater control. Otherwise, why would I invest money into solving the problems of other countries, when I can solve the problems of my own country? Why should I invest a greater amount of resources for less control of those resources? Small countries who are in SWORD hence won't have as much of a say in what happens. If everyone had equal control, then if you upset a larger country, they could decide to stop contributing which will significantly weaken SWORD because they contribute more. If power is equal to contribution, a small contributor will likely be ignored in favor of a big contributor.
Your next problem is going to be how to stop World War III. A world war isn't going to be caused by shadow organisations doing dodge james bond or mission impossible stunts. World wars are funded by countries and super powers. The same countries and super powers which are funding SWORD. Look at the current state of the UN. If a country wants to push an agenda, lets say, remove all nuclear bombs from America, they are going to be pushed back by America due to the funding problem above. If you get the support of other powers, like Russia or China, and force the resolution onto America, America will likely just leave and prepare all their nukes for a MAD scenario and just be like..."I dare you try and stop us". Likewise, similar situations are going to happen with scenarios that are in the interests or favor of a super power. For example, Russia annexing Crimea. SWORD won't stop it, because Russia and its allies are going to veto anything you do because you are acting against their interests. You could send in SWORD, but then you have World War III. Russia (and its allies) vs SWORD (the rest of the world).
Then there is the problem with a country giving military control to SWORD. No country would allow SWORD to control their military. The military will put their nation above the interests of a global organisation because their country is where they live, where their families live, the person who pays them and so on and so forth. Military personal will hold sensitive information important to their country and will be very resistant to sharing any of these details. Like literally imagine the NSA willingly telling everyone that they are spying on you. Its just not going to be happen. Likewise, sensitive information like Nuke locations, missions and covert operations just won't be shared, because the interests of a country for a country is more important that the interests of the world or your super powers. Like... as if North Korea is going to dismantle its nukes just because its part of SWORD.
That leads into secret military bases. Its hard to keep a military base secret. Think of Area 51. If you keep people out, people will know about it. If you kill them all, people go missing and their relatives will know about it. At best you can stop them from finding out whats inside, but they will know its location (especially with satellite images). Also a country isn't going to just allow a military base in their borders if its not an ally, providing them with resources they need or supporting them in the first place. Look at the China America situation in South Korea. There is no way China would allow a base which America contributes to and has sway over in their borders, because it will threaten their national security. It doesn't matter why you want the base, it is a potential threat to them and they will do everything they can to stop it (If I recall, they told their citizens to not go to South Korea, and South Koreas tourism industry took a huge hit).
Finally, You can't have a secret and global military force that is well stock piled and has access to top of the line technology. It might of worked at the start of the information age, but its not going to work any more. The global research community for example will know about the top ongoing projects. Number 1 scientists in their field don't just disappear for no reason. Satellite images are going to reveal base locations and the shear number of resources and services the military require means that at least 1 person in the entire world who supports this will blab about it, and thats all you need.
So if SWORD did exist, it wouldn't do much at all because everything it does is going to act against the interests of the super powers who control it. It would basically be a shared army of countries who have similar goals like NATO.