tl;dr: I decided not to do it.

Since Enzyme is obsolete as of React 18, you would use <render /> and not <mount />. Therefore, you don't have a wrapper to manipulate and observe the state of the component being tested.

Furthermore, I'm really only testing the component because it's a best practice, and because of the expectation of future complexity. The state change itself is trivial:

  useEffect(() => {
    if (!jwtToken) { return; }
    fetch(apiRouter.periodsPath()).then(r => r.json()
      ).then(inns => {
        setPeriodsList(inns)
    })
  }, [ jwtToken ])

Does this really need to be tested? I decided that it isn't. Again: front-end testing is important and may be critical, but only for things that actually matter. 

It makes no sense to bog a developer down out of principle.

The test code that I was trying to get working, was as follows:

const spy = jest.spyOn(window, 'fetch'
  ).mockImplementation(mockFetch);
render(
  <AppProvider jwtToken="some-token" >
    <PeriodsProvider periodsModalOpen={true} >
      <PeriodsModal />
    </PeriodsProvider>
  </AppProvider>
)
expect(spy).toHaveBeenCalled();

The mockImplementation, for the curious, was as follows:

const mockFetch = async (url) => {
  logg(url, 'mockFetch')

  if (url == `${config.apiOrigin}/periods.json?jwt_token=some-token`) {
    return {
      ok: true,
      status: 200,
      json: async () => ({ hello: 'world' }),
    }
  }
}

logg() is a custom logger. 

Now, however, I no longer use a spy method and no longer fetch in the test:

test('PeriodsModal', async () => {
  render(
    <PeriodsCtx.Provider value={{
      periodsList: [],
      periodsModalOpen: true,
    }} >
      <PeriodsModal />
    </PeriodsCtx.Provider>
  )
  const el = screen.getAllByText("Some Text")[0]
  await expect( el ).toBeInTheDocument()
});

So, I force the modal to be open so that the element is visible. And the useEffect, fetch event doesn't happen, because the context (and provider) are given in the text - they replace what's in production, so the fetch doesn't happen in test.

Voila! I both (1) have a functioning test, and (2) didn't get bogged down in a complicated technicality at this time. This additionally serves as an example of how to solve some technical questions on time allocation of the dev team.