It is possible, but I think the hatred between the governments, and the demands of the colonies and outposts, need to remain relatively modest to keep the war cold, as this scenario does have a few potential risks.
One is the asymmetry between the factions. The central government, which I presume is located on Earth, would logically have a much bigger resource base and a significantly larger and more dispersed population. Even if it has the technological ability to construct these nuclear weapons, a space station or astroid colony might not have access to enough fissile material for an arsenal large enough to ensure the destruction of the Earth government.
Conversely, Earth can construct many more missiles. And even a single hit is likely to wipe out the entire population of an outpost or colony, if not in the blast, then by damaging the many essential facilities to make life outside of earth possible. So the destruction is probably not exactly mutually assured. The war staying cold requires the central government to value the lives of Earth's population higher than their control of the colonies. Which is entirely possible, minor nuclear powers like North Korea can still deter greater nuclear powers in our world. But it requires the central government not to be too ruthless, or to perceive independent colonies as an existenial threat.
The second thing to keep in mind is detection. If a missile travels towards you at a significant fraction of the speed of light, and you lack FTL sensors or FTL communication with sensors placed further out, you may not notice the missiles in time to counter-launch your own. This makes it less likely you'll get a cold war, and more likely that one side will strike first, motivated by the Prisoner's Dillema.
Of course, the distances the missiles have to travel in an interplanetary war are also very large. So I recommend you do a little calculation when you decide exactly how big of a fraction of the speed of light your missiles can travel, and from what distance missiles can be detected. Keep in mind that a missile could turn its engine off after accelerating, at least when striking a planet or astroid instead of a space ship, making it harder to detect.
Also, since your proposed cold war has more than two sides, it's not enough that governments can see an incoming missile. If they can't reliably tell which of their opponents launched a particular missile, that undermines the deterence that would lead to a cold war. The Martian People's Republic might decide that, rather than dealing with a cold war against the Jupiter United Spacesations and the Earth Federation, they might position a space ship in between Earth and Jupiter, launch some missiles at one of them, and break out the popcorn as their rivals kill each other off. Unless you'd like that kind of plot point occuring, you'll want your setting's sensors to be good and either long ranged enough or spread out throughout the solar system by all parties, to prevent such shenanigans from working.
Edit based on comments
I presumed that the Nuclear Shaped Charge and Bomb Pumped X ray laser were weapons that were mounted as warheads on top of missiles traveling at 3% of the speed of light, until they were close to the target. From the OPs comments, I understand they inteded the weapons to be build and fired directly at enemy targets, without a missile system.
The Nuclear shaped articles states an optimal Casada Howitzer would be "able to extend the range of small nuclear pulse weapons to useful distances (in the thousands of kilometers)." The Bomb pumped laser notes that at 10,000 km, the blast is too difuse to damage an ICBM. Both of these weapons are, in terms of interplanetary travel, short range weapons.
This makes for a different dynamic than our cold war. It means colonies can slap together these weapons, declare independence, and tell Earth that if they send a ship full of marines to land near or dock with the colony to conquer it, they'll use their weapons and blow up the ship. That is a deterent, perhaps enough of one that Earth decides it's not worth the effort to try and retake the colony.
But unlike our cold war, the colony's weapons pose no threat to other colonies or planets, or even any ships that keep their distance. The threat is only to those (military) assets that one side decides to send at their opponent's territory. It's more like World War 2's western front: The Germans had weapons (bunker mounted cannons) that could shoot at ships trying to land troops, whose projectiles couldn't be intercepted. It was a deterence (along with the other assorted problems of an amphibious invasion), but the Allies decided they really wanted to invade, so they spend quite some time preparing their invasion in relative safety, before attacking with enough ships to overcome the defenses.
This seems the more likely scenario, unless Earth cares little about retaking the colonies (in which case, there probably wouldn't be a cold war, just international relationships between independent states). Earth can take its time, scout where the colonies' weapon platforms are located, and when they feel ready send several transports, and escorts with those same weapons to try and attack the colonies' weapons before they fire.
Or, if likely cost of such an invasion is too high, they could go for the WW1 solution and blockade the colony with warships patrolling more than 100.000 km away, well outside the weapon's ranges, but close enough to intercept any departing space ship before it gets anywhere.
This latter option may be your best bet for a cold-war like scenario. If space ships are very expensive while the weapons can be easily mass produced, several governments who strongly dislike each other may decide not to openly attack one another, because each of them are too heavily defended to make an attack cost effective. But outside of the range of the planet or colonies' defensive emplacement, limited conflicts are possible.
You do lose the cold war's existenial fear though, since the opponent's weapons are only a threat to any ships send right to their doorstep. There's no threat that a push on the red button could kill everyone. I don't know if that is important to have for your setting.