I agree with Kushagra, “There will be an explosion, but a conventional one, not nuclear. Nukes don’t arm themselves, before they have hit the target, and since it is shot midair by a missile, it has not hit the target yet.”
An anti-ballistic missile (ABM) is a surface-to-air missile designed to counter ballistic missiles (see missile defense). Ballistic missiles are used to deliver nuclear, chemical, biological or conventional warheads in a ballistic flight trajectory.
Interception is not about the payload (nuclear, conventional, or chem/bio, EMP, etc) but about the kind of missile.
The term "anti-ballistic missile" is a generic term conveying a system designed to intercept and destroy any type of ballistic threat, however it is commonly used for systems specifically designed to counter intercontinental ballistic missiles (ICBMs).
Now if I understand the reason for your question…
There are only two systems in the world that can intercept ICBMs. Besides them, many smaller systems exist (tactical ABMs), that generally cannot intercept intercontinental strategic missiles, even if within range—an incoming ICBM simply moves too fast for these systems.
The Russian A-35 anti-ballistic missile system, used for the defense of Moscow. The currently active system is called A-135. The system uses Gorgon and Gazelle missiles with nuclear warheads to intercept incoming ICBMs.
The U.S. Ground-Based Midcourse Defense (GMD; previously known as National Missile Defense – NMD) system. Instead of using an explosive charge, it launches a kinetic projectile.
India has an active ABM development effort using indigenously developed and integrated radars, and indigenous missiles.
India became the fourth nation in the world after United States, Russia, and Israel to acquire such a capability and the third nation to acquire it using in-house research and development.
In 2006, India successfully conducted an exercise in which the PADE (an anti-ballistic missile, called the Prithvi Air Defense , an exo-atmospheric (outside the atmosphere) interceptor system, intercepted a Prithvi-II ballistic missile. The PAD missile has the secondary stage of the Prithvi missile and can reach altitude of 80 km (50 mi).
In 2007, the Advanced Air Defence (AAD) missile system was tested successfully. This missile is an Endoatmospheric interceptor with an altitude of 30 km (19 mi).
The DRDO has developed a new Prithvi interceptor missile code named PDV. The first PDV was successfully test fired on 27 April 2014. The PDV is designed to take out the target missile at altitudes above 150 km (93 mi).
In 2016 India successfully launched advanced Defense interceptor missile named Ashvin interceptor missile from Abdul Kalam Island from Orissa coast.
According to scientist V K Saraswat of DRDO, the missiles will work in tandem to ensure a hit probability of 99.8 percent.
Would there be a detonation when intercepted?
Of sorts, yes. Setting off a nuclear bomb is a bit trickier than simply "make an explosion next to this material and it goes mega-boom." In the case of nuclear missile interception, it's unlikely that there would be a nuclear detonation. Things have to work together pretty carefully to cause a nuclear explosion.
There's some careful timing and placement of the trigger explosives in order to make the nuclear material go critical, and upsetting that timing by hitting it with another missile has a decent chance of making the bomb a lot less effective.
It'll still blow up, its likely, you'd just spread nuclear material around the atmosphere. Not good, certainly, but no worse than what would happen if there was a nuclear detonation.
The consequences of a Nuclear detonation
Nuclear electromagnetic pulse: Setting off a bomb high in the atmosphere above a city, too high for the explosion to do any damage, would STILL create an EMP that would destroy anything and everything electronic.
Goodbye all communication networks, all lighting, all electronic security systems, all computers, and all of anything that uses a microprocessor to work (including cars, which use a microprocessor for timing and all sorts of other stuff). All broken. New local stone age until everything gets replaced.
DON’T want to sound PESSIMISTIC
However, the point is, no missiles are even necessary. The enemy can just smuggle in weapons to targets. We cant even stop terror attacks. You think we can stop a few bombs being smuggled in?
Lastly, the enemy would not even need a nuclear weapon. Each nuclear power reactor has the radiation of thousands of nuclear bombs, and they can just find a way to blow some of them up.
https://www.quora.com/What-would-happen-if-a-nuclear-bomb-carrying-missile-is-intercepted-and-destroyed-in-mid-air