In Frederick Forsyth's bestselling thriller The Odessa File, a tightly knit, top-secret organisation of SS veterans arranges escape routes for former Nazis, places them in powerful positions across the globe, and develops rocket facilities in Egypt to destroy Israel with a massive onslaught of biological weapons. Typically, the story was far from being a complete invention. Forsyth took as his starting-point the revelation by the Austrian Nazi-hunter Simon Wiesenthal that a secret conference of senior SS officers and leading Nazi industrialists such as Emil Kirdorf, Fritz Thyssen and Gustav Krupp had taken place in a Strasbourg hotel on 10 August 1944 to ensure the continuation of Nazism after its coming defeat by setting up and funding what Wiesenthal called "the greatest fugitive organisation in world history".

Known as Odessa, it was invoked frequently by Wiesenthal over the following years to alert the world to the scandal of Nazi murderers living with impunity in Argentina and elsewhere in Latin America. Some, such as Adolf Eichmann, were eventually caught and put on trial. Most, like the camp doctor and author of gruesome medical experiments on prisoners at Auschwitz, Josef Mengele, were not.

In reality, as this book demonstrates, Odessa never existed. Of the key participants in the supposed meeting in Strasbourg, Kirdorf had died in 1938, Krupp, partially paralysed by a stroke, had handed over all his business interests to his son, and Thyssen, having broken with the Nazis, was incarcerated in a concentration camp. The truth about Nazis on the run, as Gerald Steinacher shows, was a lot more prosaic than the myth created by Wiesenthal.

Western Europe in 1945-46 was in a state of almost complete chaos. There were tens of millions of German and other refugees, former slave labourers of the Nazis, German and Austrian prisoners of war, and people fleeing the advance of communism in eastern Europe  most of them without papers, milling around in total confusion. In this situation, it was not difficult for exSS men to change their identities and obtain false passports. To get away from Germany, the best route lay across the Alps, over the border to Italy, and there was no shortage of experienced smugglers of other kinds of contraband to arrange their passage.

Even better for the SS men, the way lay through the long-disputed territory of the South Tyrol, German-speaking but since the first world war under Italian rule. Many of its inhabitants had supported the Germans and helped them during the brief episode of Nazi control towards the end of the war. They were all too willing to assist men they regarded as fellow Germans. Here too, it was possible to make contact with former comrades and Nazi officials who would smooth the way.

Once in Italy, the escapees found many different agencies where help could be provided, and the Red Cross was willing to issue travel documents without too many questions asked. Dedicated to an extremely broad principle of humanitarian aid, the Red Cross provided help to Jews seeking to emigrate illegally to Palestine, at the same time as giving ex-Nazis the identity papers they so badly needed. The problems confronting the Red Cross were too overwhelming for careful discrimination to be made. But it is hard to disagree with the author's conclusion that it was more careless than it should have been.

There were criminal and corrupt Italians who were prepared to forge papers for the SS men  and if the price could not be paid, there were always those who would act out of ideological conviction. Most important, there was a group of Vatican priests around the Austrian bishop Alois Hudal, the deeply antisemitic author in the mid-1930s of a tract called The Foundations of National Socialism, which he had presented to Hitler. A number of fugitives succeeded in obtaining "denazification through conversion", as German Protestant or Nazi-style "deists" had themselves baptised to get the church's help.

Pope Pius XII, a friend of Hudal's, turned a blind eye to his activities and interceded repeatedly for ex-Nazis. When a US state department report pointed the finger at Hudal in 1984, the Vatican reacted with fury, but after Steinacher's book there should be no doubt of Rome's complicity, despite the regrettable fact that the Vatican archives are still closed for this period. No wonder the current pope has put the canonisation of his wartime predecessor on hold.

However, a more important part was played by the Americans, who saw in former SS technicians, spies and experts sources of information and possible assistance in the emerging cold war. The rocket engineer Wernher von Braun, who had employed forced labourers under deadly and degrading conditions in his underground V-2 factory, was put in charge of the American rocket programme; the Nazi spymaster Reinhard Gehlen, who claimed his network of spies in eastern Europe was still in place, was rescued by the Americans and subsequently put in charge of the West German intelligence service.

Within the Vatican and the US government, significant groups and individuals helped ex-Nazis to escape justice because they saw in them men whose hatred of communism was as great as their own. After the Nuremberg war crimes trial, governments decided that strengthening German self-respect in the face of the Soviet threat was more important than meting out justice. Most surviving victims of the Nazis were too busy putting their lives back together for moral reflection on the recent past. Germans, Austrians and Italians simply wanted to forget. Nazi escapees took advantage of this growing silence and amnesia to make their escape.

And in Juan Peron's Argentina, desperate for technical knowhow and not unsympathetic to Nazism, they found a government so keen to have them that it sent recruiting agents to Italy to persuade them to come. Like all the other institutions that helped former SS men such as Eichmann get away, the Peron government was well aware of the crimes they had committed. More than half a century later, their moral indifference, as Steinacher notes, is stunning.

In this murky world of hidden identities, deception and secrets, it is good to have a book as level-headed as this one, based on thorough research. All the more pity, then, that it is written in such poor English and so shoddily edited that it does the reputation of Oxford University Press's New York office no good at all.

Richard J Evans's The Third Reich at War is published by Penguin.