If a single country used nukes at the start of a great war, everyone who has access to nukes would follow suit. Though i don't think england would be the first one to do this and hopefully no country would be stupid enough to initiate a nuclear attack. A nuclear warfare would mean almost extinction level disaster for everyone.
I guess the most powerful nuclear weapons have a blast radius of less than 100km. So a nuclear warfare doesn't automatically mean the end of all forms of life on this planet, at least for the short term. But in the long term, survival will be difficult, provided how quickly the radiation spreads through air and water.
This is only a theory. The Japanese survived after the nuclear bombings, and Chernobyl disaster. There are many people who never left the area of infection and safely live there still. Animals are also great feeling in there. The effect of radiation is still too poorly understood to speak of the destruction on the planet.
Hiroshima and Nagasaki were rebuild after a few years and hundreds of thousands of people are living there now. It seems like the radiation lasted only a few months, or years. And in Chernobyl, the blast radius was quite small. I really doubt whether it was greater than 100 meters. But the radiation was more spread-out. It even caused a spike in cancer incidence in the Scandinavia.
Hiroshima and Nagasaki could be rebuild, because all the rubble - which was radioactive, was removed. Those devastating atoms bind themselves to any sort of matter, so while you are right - air is poisonous for a few months, buildings and environment will stay so for many years.
Even after one month in shelter, single day of exposure would sterilize you. And make you lose your body hair.
Nuclear winter is also not a science-fiction. Large volcano eruptions are able to force regional climate cooling, enough nukes would block out sun for years globally - killing fauna a flora even in the areas unscathed by explosions and dropping temperature by as much as 15 degrees celsius.