Why did none of the other Kingsmen / Statesmen assist in saving the world?

Much like Tetsujin, I'm gonna have to give a meta-answer....

Because the first film was an inherently British film, that skewed British spy movies (Bond primarily, but other genre elements can be found) whilst delivering a subtle critique of the British class system.

After it found unprecedented success, Matthew Vaughn has attempted to solicit a broader audience by introducing 'American' elements, in order to more fully find favour with a US market. Many criticisms of the film stem from the notion that this element, 'The Statesmen', are reduced to mere cameos with no real traction to the plot; both during much of the sequel and as you have pointed out, any of the first film.

The Statesmen don't exist in the comics, and Vaughn likely didn't conceive their introduction during the production of the first movie. It is for this reason they effectively 'pop into existence' during the sequel, with no reference to any activity during the interim (with the exception of their rescue of Harry Hart, itself reduced to more or less a cameo).

It's disappointing that Vaughn chose not to allude to why they were unable to help during the 'doomsday' events of the first film, and I'd agree it's an oversight that raises more questions than their introduction answers.

Because they believed that both agencies might be compromised at the time.

In the first movie, the Kingsman agency is led by Arthur, who Eggsy discovers is in league with Valentine. He then goes directly to the agency and tells the only 2 people that he trusts: Merlin and Roxy.

They can't go and tell any of the other agents, because they might have been in on the plan along with Arthur. This is why Merlin entrusts Eggsy with infiltrating Valentine's base: he's the only one other than Lancelot that he can trust (because she had only just joined the agency).

The mission is also incredibly time sensitive, so they couldn't spend the time tracking down individual agents and seeing if they are trustworthy or not, they just had to get on with saving the world on their own.

Likewise in the second movie: Harry doesn't trust Whiskey and shoots him. At this point Harry didn't know Whiskey's motive for breaking the bottle with the cure in it, he might have been ordered to make sure that the cure wasn't recovered by the agency, which is why they don't tell anyone else what they're doing.