According to the wikipedia pages about India's and Pakistan's weapon's programs, each side has about 120 bombs and has tested up to the 20-40 kiloton range.
Using Nukemap to estimate damage effects from bombs of this size (all damage radii mentioned below come from this source) it seems pretty clear that Pakistan can take out the listed four cities with 120 warheads. While the arsenal isn't that impressive compared to American or Russian stockpiles, 30 bombs per city is pretty serious. Since many of my coworkers are from India or Pakistan, I'm not going to sit at my desk running simulations of nuking their homelands (might constitute hostile work environment), but we can estimate how some of these cities would fare after getting hit with this attack.
Direct Effects (Fireball, Blast, Overpressure, Thermal Pulse)
The good news is that these bombs are relatively small (compared to the 500kt - 1mt bombs fielded by the US or Russia). The radius of total destruction (fireball) is about 220m, while blast and thermal effects have a radius of 2.5km. Each attack can blanket a huge area with moderate-severe damage (30 bombs * 17 sq. km. blast area = 510 sq. km) if the bombs are dropped spread out, or a still-significant area with total devastation (30 bombs * 0.15 km^2 fireball area = 4.5 km^2) if the bombs are dropped close together, but either way, the destruction would be relatively shallow. What I mean by "shallow" is that any point more than a few kilometers from the nearest strike will be relatively unaffected.
Since New Delhi is much bigger than the moderate-severe damage area calculated above (New Delhi is about 1,500 km^2, compared to 510 calculated for moderate-severe damage), New Delhi will be horrifically damaged, but not destroyed outright. It could be like Hiroshima on a much larger scale. The other cities are much closer to the size of severe destruction, so they will be more-or-less totally destroyed, though there will be survivors and many well built buildings will still be standing.
Of course, that assumes you target 30 bombs at each city. I'm guessing a more optimal strategy would be to target by size, so as New Delhi is about as big (by area) as the other three combined, it gets half the bombs. By this method, about two-thirds of the area of New Delhi is moderately to severely damaged (weak buildings destroyed, strong buildings damaged, areas directly hit totally vaporized, unprotected people killed or injured), with the rest much less damaged. The other cities have about the same damage, though Mumbai has much, much higher population density, so the death toll there is likely by far the worst.
The Pakistani cities are much less densely populated, so the destruction will not be as bad. Karachi is like 3,500 km^2, which is just huge. Although the Indian arsenal might be larger or more powerful (this is unclear), each Pakistani city will fare better than the Indian counterpart.
I don't know the normal building materials in these cities (and most of the images I can find online are just pictures of the skylines, city centers, and tourist locations, so I can't guess at what the building materials would be in the majority of the buildings). American suburban frame houses would be totally obliterated by the blast effects (and the flying pieces would be in flames) throughout the moderate to severe damage area, while Stalinist-era concrete Russian apartment blocks would probably still be standing through a lot of the area (though human casualties would still be high, and damage would make many of the buildings unfit for habitation). The excellent US government compiled book The Effects of Nuclear war (pdf warning) has a lot of details about how this makes a big difference in destruction amounts and casualty numbers, though it doesn't compare any Indian or Pakistani cities, just Washington and Moscow. Google image search seems to show Indian cities as better built than Pakistani cities, but I have no idea how accurate this is.
Indirect Effects (Fire, Fallout)
The bad news is that direct effects aren't the only problem. Because these attacks are likely to be airburst (for maximum casualties), radioactive fallout won't be much of a problem, but fire definitely will.
Within the area of moderate to severe destruction mentioned above, almost anything flammable in the open will burst into flame. Exactly how bad the fire gets depends a lot on the composition of the city. Building materials obviously play a role. How close the buildings are to each other will impact how fast and far the fire spreads. Since the thermal effect is line-of-sight, topography could cause areas which are shadowed and therefore not ignited. Firebreaks like highways or rivers might spare sections of surviving city from the fires. Even wind direction, humidity, and how recently it has rained will play a role. That said, when something like two thirds of your city has been exposed to the thermal pulse of a nuclear bomb, you're going to have serious fires, and no amount of firefighting will make much of a difference.
These cities are pretty much doomed. Not everyone will die, there will be tens of millions of survivors, but they will no longer be functioning cities for decades.
That said, the lower population density in the Pakistani cities means they will likely fare better than their Indian counterparts, unless they are much more flammable.
Damage to surroundings
Because the bombs are fairly small, the damage outside the area scorched by the fires will be relatively minor. Many of the suburban areas of the cities will not be destroyed by the bombs, and the outlying towns and countryside will suffer no damage at all.
Damage to the Nation
Exactly how bad the damage to each nation will be is hard to estimate. I'm going to provide a few guesses in these areas: population, government, economy.
India's population losses are huge. Almost the entire population of Mumbai (because of high density), and most of the population of the other three cities has died. This is something like 51 million people, making this attack the deadliest single event in history. On the other hand, this is less than five percent of the total population. While no country has ever lost 5% of its people in a single day, many historical plagues have had much larger tolls over the course of a few months.
Pakistan's losses are somewhat less. Because of lower population density, fewer Pakistanis would die. Something like 50 million people live in the affected cities, but picking an arbitrary figure and saying the death rate is only 60% of what the Indians experienced, this is a death toll of something like 30 million. However, Pakistan has a much smaller total population, and this will constitute over 15% of the total national population. While this is still not as bad (long term at least) as some historical plagues, this is a proportion three times larger than India experienced. In population terms, I think Pakistan loses this war.
=== Government ===
I don't have any idea what will happen here, but there are some things worth thinking about.
India has lost its national capital, and several of its regional capitals as well. I am not from India, so I cannot say what that would do to the national spirit, but I think national services could be seriously damaged, and separatist movements might spring up everywhere (and be strengthened where they already exist).
Pakistan has not lost its national capital, but it has lost regional capitals. All this damage can't be good for national stability (not that great already by international standards), but it might be better off from a government standpoint than India is.
=== Economy ===
India has lost four cities out of its approximately 50 cities with populations over 1 million (8%). Based on Wikipedia numbers, it's kind of hard to guess the economic output of these cities, but I see a figure of 6% for Mumbai, somewhat more for Delhi, and less for the other two. Wild guess here: Something like 20% of the economic power of India was in these cities, and has been destroyed.
Pakistan has (mostly) lost four cities out of its approximately 15 cities over 1 million (26%). Karachi accounts for more than half of Pakistan's tax revenue, 25% of its domestic revenue, and 30% of its manufacturing sector. Wikipedia notes that Lahore is almost exactly half of Karachi, accounting for 12% of the national economy. The other two cities are about 4% and 3% each. Something like 44% of the economic output of Pakistan (and much, much more than half its tax revenue) comes from these four cities. If the strikes were targeted to maximize damage to important industries, almost all of this might be destroyed. Pakistan definitely loses on this metric.
== Effects on the world ==
The world would be pretty seriously impacted. Two of the world's five largest cities have been destroyed, millions killed, and a huge percentage of the GDP of two nations has been destroyed, with the rest of the economy in these nations likely tanking or turning inward towards rebuilding. Nations with direct ties to the two countries are hit hardest, while others are impacted less directly. I don't know the total results, but it would be bad. My guess is that it would be at least as bad as the 2008-2009 global financial crisis, and maybe far worse.
Anyway, the results would be horrific. Everybody loses, Pakistan loses more.