Can any noun ending in -ism be swapped for -ist?

It does seem that your proposition is by and large correct. One can think of a variety of exceptions, but it generally seems to hold that where you find an ...ism, there will usually be an ...ist!

But not all followers of something are ...ists, some are ...ites e.g Luddites (I've never heard Luddism used, though).

Adherents of Nazism are plain Nazis. That is a clear exception of an ...ism, without an ...ist. I have no doubt that there are others and people will think of them.

In British politics you have had, at different times Benthamites, Brownites, Blairites, Bennites etc. But I am not sure if any were ...isms.

Of course it doesn't work the other way round. There are many ...ists who do not follow an ...ism e.g. economist, sociologist etc.

No - counter example:

Catholicism => Catholic

but not

Catholicist