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You seems to want speed calculations instead of the usual common sense explanation that one shouldn't use nuclear explosions to propel a projectile.... so, let's start, I guess.

First, we have conservation of momentum: $mathbf p_s + mathbf p_p = 0$, assuming we are in the reference frame that the ship was initially at rest. Second, we have conservation of energy:
$$
U = K_S + K_P + Q
$$

Where $U$ is the total energy of the bomb, $K_P$ is the kinetic energy of the projectile, and $K_S$ the kinetic energy of the ship (or the total sum of kinetic energy of all the wreckage left by the ship). And, lastly, $Q$ is the thermal energy.

Now, it is important to notice that these two equations are valid for any and all given times. Because of that, we'll take a time where the projectile is launched, but the nuclear exposion didn't have enough time to completely destroy the ship itself (by too much). Thus, the ship is still in one piece, and thus, we are talking about a two-body system. That will greatly simplify everything.

Let $v_p, v_s$ the velocity of projectile and ship. By momentum conservation, we know:
$$
m_s v_s = -m_p v_p
quadimpliesquad
v_s = -frac{m_p}{m_s}v_p
$$

Thus, the energy:
$$
K =
frac{1}{2}m_s v_s^2 + frac{1}{2}m_p v_p^2 =
frac{1}{2}m_sleft(-frac{m_p}{m_s}v_pright)^2 + frac{1}{2}m_p^2
$$

Thus:
$$
K = frac{1}{2}left(m_s + frac{m_p^2}{m_s}right)v_p^2
= frac{1}{2}m_sleft[1 + left(frac{m_p}{m_s}v_pright)^2right]
$$

There, the projectile velocity is simply:
$$
v_p = frac{m_s}{m_p}sqrt{frac{2K}{m_s} - 1}
$$

Where $U = Q + K$. Now it would be a matter of estimating the ratio between kinetic and total energy available, or, the efficiency $eta = K/U$ of your transformation between bomb's energy to kinetic energy.

Plugging numbers: If $U = 25MT = 1.046cdot 10^{17}J$, setting $eta = 0.1%$, this means $K = 1.046cdot 10^{14}J$. If $m_s = 10^9 Kg$ and $m_p = 10^3 Kg$, then: $v_p = 0.48245c$. Almost close to speed of light. Be aware that all of this ignores the obvious problems of having a nuclear explosion to propel a bullet: if you have those problems solved, then this calculations gives you a rough speed estimate of the projectile. And keep in mind I completely guessed $eta$ (such value will depend on the actual mechanism you have to transform the bomb's energy to kinetic energy). Besides, $eta$ could be muuuch MUUUCH less than the guessed value (it probably is).

Tim's right, but let's have some fun explaining why.

Assumptions

• When you said "railgun" what you meant was "slug thrower." A railgun is something specific, and you're not describing it.




• The "gun" can't be damaged by the nuke. This is an experiment, after all.




• Neither can the shell.




• The practicality of the one-use-incredibly-expensive-hard-to-aim gun is ignored.




• The "gun" has a mass at least 1,000X of the shell. Thus, 99.9% of the final velocity belongs to the shell, while only 0.1% of the final velocity belongs to the gun (which enters a decaying orbit, devastating the southern continent of your world, but that's a story for another day). Anyway, what this does is let me reasonably assume all the energy is used to push the shell.




• Here's the whopper: we're going to assume that when they measure the explosive capability of a nuke in terms of tons of TNT, they really mean it (and the angels weep over the magnitude of this lie... but we'll get to that).

The equation of choice is F=mA. Let's be outrageous.

• The charge is equivalent to the Tsar Bomba: 50 megatons or 210 peta-joules. So, F = 210x10¹⁵.


• The shell is a useful 10,000 Kg. So, m = 1x10⁴.


• A (acceleration) = 210x10¹¹ m/s².

It has no resistance (space...) and there's no more force acting upon it. So our final velocity is darn close to twice the speed of light.

Which obviously can't happen

And that's why Tim's answer is correct: puff of light and heat.

OK, what's the problem?

A whomping large amount of a nuke's energy has nothing at all to do with kinetic energy. It's radiation. Oversimplifying something awful, you only have the mass of the charge to work with in terms of expanding gas (kinetic force) and a lot (if not all) of it just got converted to pure energy.

Most of the concussive force of the nuclear explosions we're familiar with are (a) the conversion of surrounding mass (air, soil, houses, people) into expanding plasma and (b) the shock created when the vacuum caused by the explosion collapses.

Which is a fancy way of saying you don't actually have that much to work with. Since you only have whatever convert-to-plasma materials are in the nuke itself (gun and shell can't be damaged by the nuke!), your gun will glow white hot¹ while the equally white hot shell kinda blurps out of the muzzle and lazily meanders away.

If I got this wrong, please correct me!

Edit:

Physicist137 pointed out the flaw in my math, and I'm grateful! I had fun, but go upvote his math! I'm still happy with my conclusions, but his ~0.5c answer is a whole lot more realistic than my 2c analysis!

_{¹ And it'll probably stay white hot for decades. Space is a terrible place to dump heat.}

There's a simple problem with this: shock and vibe.

Your projectile has to travel a very long distance before hitting. Your specification of "around the limit of detection by IR" is sufficient to be quite certain you will never hit your enemy with a dumb slug. You simply can't aim the gun that accurately. This is doubly true for the fact that you're about to set off a nuclear explosion (probably well more than doubly true).

This means you're certainly using a smart weapon, with onboard propellants, sensors, and computers, much like we see in all of our own ballistic defense programs on planet Earth.

So here's the real issue. Launching things fast (space fast) is hard. You have to accelerate rather quickly. Shock and vibe issues during launch are astonishingly difficult to deal with. The harder you accelerate, the nastier the requirements. Navy rail guns face serious challenges in hardening smart rounds so that it survives being fired (tests are typically done with dumb Tungsten slugs today, but it is recognized that they will need to be guided eventually to strike distant targets). You have invented a new level of "difficult launch conditions." Railguns can maintain a maximized shock and vibe for the length of the rail far better than a simple nuclear gun can.

Launch Conditions: Must survive being at ground zero of a 25MT thermonuclear bomb detonation, in a confined space designed to maximize the impact of the thermonuclear bomb on the projectile.

Ouch!

What you really want to do is stick to railguns. If you design a projectile that can survive the forces of a nuclear bomb, you immediately put that same projectile into a railgun, and get more performance out of it (railguns can accelerate linearly along the entire length of the rails, while this gun peters out as the pressure wave weakens)

Your single use ship will disappear in a puff of light and heat.

The problem with what you're describing is that a nuclear bomb is exactly the wrong type of energy source for a rail gun as it generates thermal heat and radiation, not kinetic energy, in space.

We've all read about the blast waves off nuclear explosions and how powerful they are; there is even some research into this that shows that blast (kinetic) energy comprises roughly half the energy released in a nuclear explosion. The trouble is, that blast energy is essentially the atmosphere super heating from the thermal energy released, and therefore rapidly increasing in pressure in an unconfined space.

Nuclear bombs are not designed to release kinetic energy; they're designed to release thermal energy. This can still work because of a technique known as Ablative propulsion, which amounts to superheating a surface of an object to a point where the surface becomes a plasma that pushes the object rapidly in the opposite direction to that in which the heated surface is facing.

This kind of thinking about propulsion seems to be behind Nuclear Pulse Propulsion theoretical engine systems like that originally planned for Project Orion. My reading on this to date seems to indicate that the plan was to detonate nuclear bombs outside the ship and have the blast hit a pusher plate on the back of the ship, pushing it in that direction. What is unclear in all those mainstream articles is how that blast wave is generated but the best I can come up with is that the mass of the bomb itself would become a superheated plasma that would push against the rear plate of the ship. The closest I could come up with in terms of reference to how nuclear energy can be converted to kinetic energy in space was a reference to Electrical v Direct Thrust on fusion drives, but even it talks about the plasma having to be contained to sustain the reaction which to my mind seems contradictory.

If this isn't the thinking behind nuclear pulse theory and you know what is, leave it in comments and I'll integrate it into my answer with full credits. But back to the matter at hand...

In space, there's no atmosphere to heat up, so no localised pressure differential, therefore no blast wave. What WILL happen is that the heat energy will essentially vapourise your ship and the projectile through heating them all equally.

To be fair, this will result in the aforementioned ablative propulsion for your rail projectile; as the weak point in the containment, it's going to break off (or out) first but without running some serious math I have no idea just how quickly it will escape the barrel, how much of it will be left to hit the target, or even if it would escape the barrel at all before the thermal energy expands it (and contracts the barrel) before it escapes.

Before anyone says it, yes, the barrel will constrict. As the metal expands, the inner metal of the barrel also expands, meaning that the hole gets smaller through swelling.

Guns like cannon, pistols, rifles and rail guns work through the kinetic energy of expanding gases. The idea is that gunpowder, when burnt, releases a massive amount of hot gas into a confined area, forcing out the weakest point of the containment chamber, which just happens to be the bullet. This projects it suddenly in the direction of your choosing. Yes, burnt gunpowder is hot, but it's not the heat that shoots out the bullet; it's the expanding gases.

In space, your setup has no atmosphere around it to expand, and even if it did, the kinetic energy transferred to it only represents about half the energy that a nuclear device can produce, which makes it an inefficient energy source for what you're trying to do, and would require such a sturdy, heavy ship to withstand the vapourisation long enough to transfer the bulk of the energy to the projectile so as to make it uneconomic as an investment.

For my money, you'd be far better off investing in a nuclear powerplant that feeds electrical capacitors that are capable of releasing a sudden energy burst of electricity to propel your bullet. Far more efficient process of converting thermal to kinetic energy (via temporarily being electricity), reusable and less dangerous to the rest of your fleet by not creating debris fields with every shot.