I would like to post it in the comment sections, but it'd eat up too much space, so here we go:
Losing our intelligence would be easy: a full scale nuclear war would turn the last survivors of humankind into scattered, helpless small groups too busy to scavenge food to care about education. As time passes and more people die and diversity wanes, those few who remain will be inbreeding and giving brith to a new generation plagued by growing genetical defects, among them idiocy.
Well, of course at this point, chimps should need to evolve fairly quick to take our place...but whoops, there is no time at all. For us to become who we are now, it took about 50,000 years if not more, and during this long, long walk we had a living, thriving ecosystems at disposal.
After WWIII, the world would be a monstrously cold, unforgiving, dark, radiation-filled place where the ecosystem is no more. Chimps die by collapse of food chain. We die for the same reason and the rapid and traumatic loss of infrastructures.
Your scenario, thus, cannot happen. You need another background for a starting point that will see us becoming idiots and chimps the rulers of the world
EDIT: Please look up this wikipedia article about nuclear winter.
2007 study on global nuclear war
A study published in the Journal of Geophysical Research in July
2007, titled "Nuclear winter revisited with a modern climate
model and current nuclear arsenals: Still catastrophic
consequences", used current climate models to look at the
consequences of a global nuclear war involving most or all of the
world's current nuclear arsenals (which the authors judged to be one
similar to the size of the world's arsenals twenty years earlier). The
authors used a global circulation model, ModelE from the NASA Goddard
Institute for Space Studies, which they noted "has been tested
extensively in global warming experiments and to examine the effects
of volcanic eruptions on climate." The model was used to investigate
the effects of a war involving the entire current global nuclear
arsenal, projected to release about 150 Tg of smoke into the
atmosphere, as well as a war involving about one third of the current
nuclear arsenal, projected to release about 50 Tg of smoke. In the 150
Tg case they found that:
A global average surface cooling of −7 °C to −8 °C persists for years, and after a decade the cooling is still −4 °C (Fig. 2).
Considering that the global average cooling at the depth of the last
ice age 18,000 yr ago was about −5 °C, this would be a climate change
unprecedented in speed and amplitude in the history of the human race.
The temperature changes are largest over land … Cooling of more than
−20 °C occurs over large areas of North America and of more than −30
°C over much of Eurasia, including all agricultural regions.
In addition, they found that this cooling caused a weakening of the
global hydrological cycle, reducing global precipitation by about 45%.
As for the 50 Tg case involving one third of current nuclear arsenals,
they said that the simulation "produced climate responses very similar
to those for the 150 Tg case, but with about half the amplitude," but
that "the time scale of response is about the same." They did not
discuss the implications for agriculture in depth, but noted that a
1986 study which assumed no food production for a year projected that
"most of the people on the planet would run out of food and starve to
death by then" and commented that their own results show that, "This
period of no food production needs to be extended by many years,
making the impacts of nuclear winter even worse than previously
In 2014, Michael J. Mills (at the US National Center for Atmospheric
Research, NCAR) et al. published "Multi-decadal global cooling and
unprecedented ozone loss following a regional nuclear conflict" in the
journal Earth's Future. The authors used computational models
developed by NCAR to simulate the climatic effects of a regional
nuclear war in which 100 "small" (15 Kt) weapons are detonated over
cities. They concluded that:
global ozone losses of 20–50% over populated areas, levels unprecedented in human history, would accompany the coldest average
surface temperatures in the last 1000 years. We calculate summer
enhancements in UV indices of 30–80% over Mid-Latitudes, suggesting
widespread damage to human health, agriculture, and terrestrial and
aquatic ecosystems. Killing frosts would reduce growing seasons by
10–40 days per year for 5 years. Surface temperatures would be reduced
for more than 25 years, due to thermal inertia and albedo effects in
the ocean and expanded sea ice. The combined cooling and enhanced UV
would put significant pressures on global food supplies and could
trigger a global nuclear famine.
You are severely underestimating the effects of a nuclear war here...