As others have said, the book does a far better job explaining how what would otherwise seem a clear-cut case of nuclear terrorism becomes a major incident that nearly turns into WW3. As others have said, a lot of it has to do with incompetent officials on both sides essentially bungling things until they spiral out of control.
In the book, the nuke is Israeli largely to explain how the villains got it - rather than the cartoonish neo-Nazi caricatures seen in the film, they are an Islamic fundamentalist faction of the PFLP based in Syria. Incensed by a new Israeli-Palestinian peace plan, they come into possession of the lost Israeli nuke from a Druze farmer who found it (as in the film, it came loose after the Israeli A-4 carrying it was shot down in 1973, landing away from the wreckage and becoming buried in the desert). They ally up with some old Red Army Faction friends from East Germany, themselves angered by the Soviet "abandonment" of Eastern European communism, as well as a radical Native American nationalist who helps them smuggle it into the US (that last one was stretching it a bit IMO).
The US actually does assume that it's a lost Russian nuke at first. The book is set ca. 1990 - 1991, as the USSR is withdrawing from Eastern Europe and clearly on it’s last legs itself (it was actually published just days before the August Coup). US intelligence hears rumors that Soviet forces lost track of a few tactical warheads in the process of withdrawing from East Berlin. Meanwhile, fed faulty intelligence by the Soviet Vice President (a CIA asset who, strangely enough, is written as a Yeltsin expy and is using his position in an attempt to seize power for himself) the US comes to believe that hardliners in the KGB and the Red Army are plotting to overthrow not-Gorbachev and bring back the bad old days.
The reason the nuke is assumed to be Soviet is largely due to National Security Advisor Liz Elliot's influence on the President (she is his lover as well as his advisor), as well as further terrorist machinations (the East Germans are able to set off a firefight between US and Soviet troops in Berlin). Knowing about the rumors of lost Soviet nukes and coup plots in the Kremlin, she assumes that the Russian hardliners have nuked Denver in an attempt to distract the US so that they will be unable to interfere when the hardliners seize power in the USSR. Despite this making very little sense (this is lampshaded by Ryan, who points out just how little sense would make for Soviet hardliners to risk nuclear war with the USA while they are already going to have their hands full trying to overthrow the moderate government), the President trusts Elliot and distrusts Ryan thanks in part to personal animosity between the two men and Elliot's previous machinations to discredit Ryan, who she saw as a rival.
Really, I recommend reading the book. It’s not quite Clancy’s best (there are a
lot of subplots - all are tied together in the end, but a good deal of it seems unnecessary, like following a redwood tree as it is transported to Japan so that it can fall off a boat and cripple a US submarine at a critical moment, or Liz Elliot trying to get Ryan fired by spreading rumors of an affair leading to a near catfight between Elliot and Ryan's wife), but it's far better than the film. While some of the changes made in the film can be excused (having to update the setting to address the collapse of the USSR and developments in post-Soviet Russia), many more are simply ridiculous. The villains were changed from Islamist terrorists to neo Nazis
explicitly as a concession to CAIR, for instance.