It depends on the genetics of the Ameraucana rooster. A word of warning. If you got him from a hatchery he is not Ameraucana but and EE. There is no telling what his genetics are.
If the rooster is not barred, you are part way to black sexlinks. All the male offspring will be barred and none of the female offspring will be barred. There could be a problem with this. If he is white, you won’t be able to see the barring.
There could be another problem. Depending on his genetic make-up, you might not be able to see the spot on the male chicks’ heads. There are plenty of genetic combinations where the chick down does not allow you to see the spot.
If the rooster is not barred, you will get roosters with barring and hens without barring. But whether or not you can see that depends on the specific genetics of the rooster.
I’m not exactly sure what you mean by “cuckoo”? To me, cuckoo means a barred chicken where the barring is jumbled up like on a Dominique or Cuckoo Marans instead of crisp and sharp like a good Barred Rock. Whether the barring is jumbled or crisp depends on whether the chicken has the fast or slow feathering gene. You may mean something different by “cuckoo”.
Assuming the rooster from that cross is really a black sex link, he will have one barred gene from his mother and one not barred gene from his father. He will randomly give one or the other to his offspring of both sexes. So about half his offspring with that blue hen will be barred and half will not.
A further complication is that I don’t know exactly what you mean by “blue”. You may be talking about a solid blue hen or one that is only partly blue. Even if she is solid blue, there are different ways to make that genetically. With the unknown genetics of your blue hen and the guaranteed mixed up genetics of that cross-breed rooster, I’m not sure what the barred chicks from your second cross would actually look like.
What I feel safe saying is that about half the chicks from that last cross should show black and half will show blue. The way the “blue” gene works, if the hen had two copies of the blue gene she would not be blue. She would be splash, a black and white mix. If she has one copy of the blue gene, she will show blue wherever she would normally show black. And if she has no copies of the blue gene, black will be black.
Since she will randomly give a blue or not blue gene to all her offspring half will be blue and half will be black.
So assuming all these things come together, what I’d expect you to get with that second cross is ¼ of the chicks will show blue barring, ¼ black barring, ¼ blue with no barring and ¼ black with no barring. But one thing I have no confidence in is where that blue or black might show up due to all the other unknown genes that could be in your mix.
There is another issue too. If your Americana rooster is a true Ameraucana he will be pure for the blue eggshell gene and will give a copy to all his offspring. The Barred Rock will give a not-blue eggshell gene to all her offspring. So their offspring will all have one blue and one not-blue eggshell gene. Which they pass to their offspring is random. So unless your blue hen is laying blue or green eggs, about half the offspring will have a copy of that gene and half won’t.
A convoluted mixed up answer I know, but you did not pick simple genetics.