The most one could logically say is that secular atheists and radical Islamists have a shared interest in wanting the Christian influence on Western society gone--but after that point, they would be at absolute loggerheads again.
That's not logical at all. Radical Islamists hate Western atheists every bit as much as they hate Western Christians, and atheists would never work with radical Islamists since they're religious fanatics. We have as much to lose from them as Christians do. If you have a problem with Christian influence on Western society, you wouldn't team up with religious fanatics and help gain them influence, making your situation fifty times worse. That makes no sense.
Their goals aren't shared at all until you enlarge them to a ridiculous degree. Suing because a judge is using the Ten Commandments as a guide in court is not the same as wanting to engage your enemy in a holy war. You might as well say the two sides would team up because they have a common interest in breathing air.
(Actually a scenario I have worked with for possible use in a fanfic...)
And that's the only arena where a marriage of such disparate interests has any chance of existing, much less succeeding.
You'd be looking at something pretty tenuous, the equivalent of both submitting an amicus brief on the same Supreme Court case because the potential ruling could favor both. In fact, the legal arena is where you would most likely see it--for a time...because eventually, if Christian influence were to be eliminated, you'd then have the inherently incompatible desires for total secularization if not banning of religious expression versus sharia law. "Tinderbox" wouldn't even begin to sum that up. It would get nasty, fast.
So, we've gone from a future secular atheist controlled government dominated by radical Islamists according to Newt to they might file an amicus brief on the same case according to you. Which begs the question of why lend any credence to Gingrich's comments at all, and not simply dismiss them as the fearmongering tactics aimed at the two minority groups his base
dislikes and distrusts the most that they are?
We're talking about people who can go from thinking Obama is part of a radical Christian church to not religious to a radical Muslim and both a fascist and a communist in the span of a single election cycle. They'll pretty much buy anything you throw at them, no matter how contradictory it is.
Since when do radical Islamists file amicus briefs in the Supreme Court, anyway? They might try to blow up the Supreme Court...
Atheists don't want to ban religious expression.

We're not the enemies of Christianity or any religion.
The same can be said of most Muslims, too (not wanting trouble or to be the cause of trouble). The problem is that fanatics can appear in any group, irrespective of religiosity or lack thereof, and when they happen to get power, things can get dangerous very fast. Human nature being what it is, this is always a threat.
There's no threat of either atheists or radical Islamists getting power here. That's just paranoid Red Dawnism.