可以考虑以下说法:
“The document you needed has been duly signed (and mailed/delievered by courier to you)”
我们中文会说 “要的签字已经签好了”, 但是英文很少这样说的。它的实际意思是文件已经签好了,所以就应该直接说明白。括号里头是看需要而定,意思是已经把文件给速递出去给对方了。工作上,光是签好文件估计是不够的,是否还得送回去?还是等对方来取?说明一下既简单,又有条理。
希望帮到你。
The signature you need is done.
The signature you required is done/signed.
The documents needed signature is done.