Define a constant CUSTOM_FORMAT = "Custom format" and use it when adding the item to combo and here. Otherwise, one can change the combo label and forget to change it here (and elsewhere, as necessary).

The proper call is self.format_cb.currentIndexChanged.emit(). But don't do this. It's better to call self.variable_changed directly, so the flow is more transparent. Also, self.variable_changed perhaps does something that shouldn't be done if the change is triggered by the user editing this line edit, in which case we'd split self.variable_changed in two functions and only call the second part here.)

By the way, I think that you can't/shouldn't emit this signal. setCurrentIndex will already emit this signal (see https://doc.qt.io/qt-6/qcombobox.html#currentIndex-prop). That is, the proper code would be

    self.variable_changed()

but this will call self.variable_changed() twice (once because of the change in combo and once because of the change here), so you should probably write

if self.format_cb.currentText() != CUSTOM_FORMAT:
    self.format_cb.setCurrentIndex(self.format_cb.count() - 1)
else:
    self.variable_changed()

Your original variable is a string. You must change its type to time variable, like this:

        editor = self.widget.findChild(VariableEditor)
        tc = editor.layout().currentWidget().findChild(QComboBox,
                                                       name="type-combo")
        tc.setCurrentIndex(3)  # time variable
        tc.activated.emit(3)

When you saw an "unchanged" date, it was because your variable remained a string, and changes that you made in the edit dialog were ignored.

I know the variable didn't change (it stayed StringVariable). I said I tried everything, we even had a look at the code with Lan, but couldn't figure out how to write a proper call.

output[0].metas[0] is a numpy.float64. (Don't ask why; output[0] is Instance, but .metas returns a numpy array. This ancient part of Orange is a mess and it's better not to use it.) A proper test would be

        var = output.domain.metas[0]
        self.assertIsInstance(var, TimeVariable)
        self.assertEqual(var.str_val(output[0].metas[0]), "2024-01-01")

The last line replaces what you tested. The assertion before that tests that variable type is indeed change; this test would fail with your previous code.

A better alternative to the last assertion is

        self.assertEqual(var.str_val(output.metas[0, 0]), "2024-01-01")

because it avoids Instance. Or, even better.

        self.assertEqual(
            output.metas[0, 0],
            datetime.strptime("2024-001", "%Y-%j").replace(tzinfo=timezone.utc).timestamp())

because it avoids turning the date into string and tests that the number representing the date matches the date from the test. This makes it easier to test a few other formats, including those that also have time (e.g. MH:MM), not just dates.

Don't forget to remove this! I suggest you remove it now, and when you're making commits, add with add -p and skip this chunk.