To be honest, I think that the Swiss will only join "the Axis" (in the hybridised, fascism-lite manner I described in my post above) in the context where there is A) no Free France, and Vichy France ceases to be "Vichy France" and is instead simply "France" and B) Britain makes an early peace because it thinks it can't win. This occurs either because of political PoD's in Britain or because some bloody stupid idea like Operation Pike had gone ahead and pushed the USSR into the war on the Axis side, putting Britain's Middle Eastern dependencies and India under a more direct threat. In such a circumstance, where there seems to be a total victory in the West and Germany is preparing to invade the Soviet Union, Switzerland might economically integrate itself with whatever form of "European Community" the Nazis and the Italians negotiate between themselves, with a slow burn smattering of authoritarian laws aimed initially at "Communists" but eventually targeting more and more groups. Democracy will not be abolished, but "reformed." Eventually, much to their privately expressed regret of course, anti-Semitic laws will have to be imposed, and political refugees from Germany and Italy repatriated to their home countries.
(As an interesting side-note, though the Swiss were generally regarded poorly by German Nationalists and Fascists, this wasn't entirely universal. Moeller van den Bruck, author of the highly influential book The Third Reich, was a fan of Swiss Democracy as he believed it represented a more "Germanic" form of democracy than Western Parliamentarianism. This might provide a model for how Nazi propaganda might explain away Switzerland's quaint little system differences from that of Greater Germany.)
This will be sold as a measure to ensure Switzerland is not invaded and partitioned, and to ensure its political system is "modernised" to match the new era of apparently victorious totalitarianism. However, even then I don't believe they wouldn't ever sign a explictly military alliance - and they will participate in the inevitable invasion of the USSR only by permitting volunteers (Swiss Volunteers OTL, who usually had dual nationality with Germany, had their citizenship taken away) as Vichy and Spain did.
(As an interesting side-note, though the Swiss were generally regarded poorly by German Nationalists and Fascists, this wasn't entirely universal. Moeller van den Bruck, author of the highly influential book The Third Reich, was a fan of Swiss Democracy as he believed it represented a more "Germanic" form of democracy than Western Parliamentarianism. This might provide a model for how Nazi propaganda might explain away Switzerland's quaint little system differences from that of Greater Germany.)
This will be sold as a measure to ensure Switzerland is not invaded and partitioned, and to ensure its political system is "modernised" to match the new era of apparently victorious totalitarianism. However, even then I don't believe they wouldn't ever sign a explictly military alliance - and they will participate in the inevitable invasion of the USSR only by permitting volunteers (Swiss Volunteers OTL, who usually had dual nationality with Germany, had their citizenship taken away) as Vichy and Spain did.