They've constructed a couple of working prototypes actually, some of them were destroyed by allied bombing though. Try googling for "Amerika bomber project" if you are interested.
http://rarehistoricalphotos.com/mes...ive-able-strike-continental-usa-germany-1942/
Working prototype of Horten Ho 229 was captured by Americans, reconstructed later in U.S., and now their B-2 bomber looks awfully similar to this old German flying wing.
Heh, with requring around 25250 litres of fuel, one has to wonder how they wanted to build enough of those to actually make a difference in the war. A handfull of bombers hiting some targets on the coast of the US, surely wouldn't have really hurt the US economy and industry hard enough. Albeit, the fact that the Germans focused to late on long range bombers, was a strategical mistake that showed it's effects when they attacked the Sovietunion. The He 117, saw a relatively low number of planes, when compared to the production number of the B-17 and B-29. around 1100 compard to the 12 000 B-17s. And the Germans lost 600 planes in a bombing raid against London in 1944 as retaliation for Hamburg, and it didn't really had much of an effect on the British war efforts. And we have not even talked about the number of british Bombers, which had their effect on the Germans as well.
Also, Germany had some spies & saboteurs in the US mainland. If they wanted to use a dirty bomb they could have just gotten the material (if they had enough of it) to those folks who would have dispersed it from the top of a skyscraper in NY or something. Or put it in the water pipes or something.
The question is, what would they gain from it? If they wanted to attack the civilian population, they could have done this with conventional means already, and probably with similar effects. Like gas or poison or just explosives in crucial installations. But the thing is,
1. It's not that easy.
2. The enemy might retaliate.
The Nazis havn't been that stupid. They knew, what ever they do, the enemy might respond the same way. It is one of the reasons why chemical weapons, like poison gas, has never been used in WW2. Even though, both the Germans and British had huge stockpiles of it, for the cases that it was required. This served as deterrend. It was even discussed in London, if they should use it against Germany, as answer to the V1 and V2 rocket attacks. But they quickly droped that idea and just increased bombing raids over Germany.
If the Nazis would have really used a dirty bomb? I doubt it. You have to remember, if the Nazis would have been capable of delivering a dirty Bomb, which wouldn't have been easy to achieve if at all, the allies could have done it as well. And most probably much easier, simply with planes droping it from the sky over German cities. A lot of ifs and buts. Even if they had some spare material, they would have never wasted it on such a high risk and low gain project. At the end of the war, they had barely enough material to get a fision reaction going in some experimental reactor. And it isn't just like you can walk around with nuclear materials or planing some really devastating and sophisticated missions, without someone to notice it. Not to mention the Nazis never had the manpower nor the required spy-network to achieve something like that.