Assuming the North has leadership with half a brain, they would have rejected any offers of foreign assistance right away. First of all, the Union didn't need foreign help at all. Whatever its problems, lack of men and material were not among them. What on Earth could any European state offer the Union that they would have needed?
Second, accepting foreign help would have give the Confederacy much greater political credibility. The North's official position was that the South was in rebellion against legitimate authority and was not an independent state, no matter what it said. Nations typically don't bring in foreign troops to suppress domestic insurrections, but often ally with other nations in war against other nations. Moreover, it would be extremely useful to Confederate propaganda efforts, just as the British use of Hessians during the Revolutionary War was a goldmine to Patriot propaganda.
Third, asking for foreign help would give the Democrats a political issue made of gold to beat the Republicans over the head with in the 1862 midterms and 1864 general election. They made great hay out of portraying the administration as incompetent and unfit to carry on the war. If the North were doing well, the Democrats would be able to say, they wouldn't have needed to ask for foreign help. Their campaign would be given an enormous boost.