Let's suppose that on a planet the intelligent beings started a nuclear war and destroyed most of the surface of the planet. Now, the intelligent beings are all extinct. Assuming this were an earth-like planet with life forms just like those on earth, what sort of life forms would evolve after the nuclear war? By this I mean life forms that evolve from the species that survived the nuclear war.
Mushrooms are probably a best bet for evolving! Mushrooms are very versatile and are decomposers, meaning they thrive by breaking things down. With the death of a dominiant species, they would surely feast.
Plus, there is a special type of mushroom called radiotrophic, meaning they can convert radiation to chemical energy! This would give them a very significant advantage
Mushrooms, exposed to radiation over time, and feasting on a slew of creatures (dominant species and whatever went down with them) could surely evolve into something interesting!
The correct answer to this question really depends on just how badly the surface of the planet was destroyed. Life is more tenacious than we generally think and if the nuclear war merely took out the cities and left most of the regional areas alone, life could survive quite well.
While not exactly a nuclear blast, Chernobyl was (at the time) seen as a massive disaster and the meltdown caused massive contamination, yet plants have thrived in the area. This has left many scientists scratching their heads, but it would appear that if the contamination in certain areas of the planet are only contaminated up to a certain threshold (we don't currently know what that threshold is) then at least some plant life is going to survive, possibly thrive.
This is very important. You can't have animal life without plant life, at least not for very long (geologically speaking).
Animals though; that's something else. Most animals are not going to cope with radiation. The mutations alone will cause all sorts of problems and many animal species can expect to go extinct, or have substantially reduced population sizes.
The exception to this rule is the cockroach. They won't survive nuclear blasts of course (the linked article explains that) but those in the regional areas should survive the subsequent radiation quite well. Many invertebrates may well fit into this category but I don't have specifics to hand.
But, let's assume for a moment that the main species that survives the holocausts is a cockroach. That means that in future eons, it's possible that the next advanced species to exist would be some form of insect.
I say possible, but it's still unlikely. Insects don't have lungs so they have a limit to their maximum size because they have to absorb O2 through the surface of their bodies, and that O2 has to reach the inner most areas of their bodies. This limit on size and O2 absorption methods also puts a limit on their brain size, meaning that human like intelligence in atmospheric proportions similar to those on Earth today is a practical impossibility for cockroaches.
The one caveat on all this is a nuclear winter. The answer above assumes that enough of the earth's surface gets at least some sunlight to support plant growth, allowing some plants to survive. If enough bombs drop and throw enough material into the sky to block it out for a year or more as many suggest would happen in a nuclear war, then all bets are off for complex life.
We'd be starting again from whatever bacterial lifeforms manage to hang on.