the problem with even a Soviet/Axis super alliance going against the USA is that they really don't get the kind of boost in the one thing they need the most, and that's on the naval side of things. Since of the powers involved in this theoretical alliance, the only one with any kind of real Naval capability is Japan, and they shot their wad pretty much within 1 year of pearl harbor. In the Atlantic the situation isn't too good looking for the bad guys, even with the Soviets added to the equation and the Brits taken out of the picture, the US is going to rip the Kreigsmarine and the Soviet Navy a new one in due time. Especially without the need to provide trans-atlantic convoy defense, the USN can devote almost all of its resources toward coastal defense which will make a huge bottleneck for your super evil-axis ground troops getting to North America, especially since their ground troops are going to suffer exceptional casualties even before setting foot on american soil as they get pounded on from the air by US carrier based aircraft, and from below from US submarines. Not to mention that the USN would get all of the planned Iowa class battleships, but also the Montana's as well which would make even the most ardent German or Russian sailor quake in his boots.
More or less the Russians/Axis would need to take the USN out of the picture in order to establish any kind of sustained foothold on North America, and in order to do that they would need to successfully take out most of the East or west coast to secure one of the oceans which in turn would be made nearly impossible by the presence of that very same Navy.
This of course is before we even get to the issue of an actual land war in the Continental US, which as has been said before would be a logistical nightmare. Granted the American public would probably not throw themselves at the invaders with the suicidal abandon that the Soviets did against the Germans, but at the same time, the US doesn't need to. The logistical chokehold will slow the invading forces down so thoroughly that geurillas won't have to do much to keep them from advancing, which of course keeps german generals from fighting a quick "lighting" style war that they are comfortable with.
Finally the issue of the A-Bomb is moot anyways, the Germans were the only nation of the four mentioned even close to developing a weapon during the 40's, and there is no guarantee that they would have actually figured it out without some help form elsewhere. I have heard a number of different things regarding the bad assumptions that the Germans had regarding what it took to actually build a bomb, one being that you needed to make a nuclear reactor go critical, and the other being that they needed like a 10 foot diameter ball of uranium or plutonium to get a working bomb. Now even if they managed to build a successful bomb, they would have had to get it to the target. They could have used one that was carried by a plane, but then again I wouldn't like the chances of a German bomber getting through US AA defences, especially with the OSS decoding German radio traffic and knowing when and where to expect such an attack. Of course there is the option of using a V-2 although I don't think that any V-2 was capable of carrying an A-bomb, and even so they would have needed a secure place to launch from which means a naval platform which would be under constant threat from the USN, or a land based site, which would be under constant threat from Geurrilla sabotage, or USAAF airstrikes. Of course once again the ability of the OSS to read enigma would make securing such a launch site quite problematical.
Of course there is always the chance that such an alliance could have succeeded in conquering the US, but it is a miniscule chance and requires an ASB level of cooperation of these powers, as well as uncharacteristically intelligent level of leadership on the part of most or all of these nations.