Bug report on webkit.org
Below is 2 form.
First one has only a simple texte input.
Second one has a file input.
On Service Worker, a header X-FROM-SW is added with the user timestamp.
On Server side, a simple JSON is returned containing this header, or false if not found.
On Chrome and Firefox, the from-sw is correctly added.
On Safari, it's added only on the simple text form.
It's not working on Uploaded form, where it doesn't go to Service Worker fetch event.

test.js file
sw.js file

Simple form

Form with upload

Form with upload, file will be removed from FormData

Workaround

Form with upload, file will be rewrited in dataUrl to be posted and recreated as regular file upload on Server side

As suggested in the third comment here, the last form read the file, transform it into a dataUrl and send it through regular POST data.
Then, on server side, the followin PHP code is used ASAP in the code in order to recreate the same behavior than regular file:

if (isset($_POST['_files'])) {
foreach ($_POST['_files'] as $name => $file) {
    list($type, $data) = explode(';', $file['data']);
    $type = substr($type, strlen('data:'));
    list(, $data) = explode(',', $data);
    $data = base64_decode($data);

    $tmpName = tempnam('/tmp', 'phpFile_');
    $handle = fopen($tmpName, 'w');
    fwrite($handle, $data);
    fclose($handle);

    $_FILES[$name] = [
        'name' => $file['name'],
        'type' => $type,
        'tmp_name' => $tmpName,
        'error' => 0,
        'size' => filesize($tmpName),
    ];
}
unset($_POST['_files']);
}

Then, when you want to move the file to another location, you'll have to use this code:

if (isset($_FILES['file'], $_FILES['file']['tmp_name'])) {
if (is_uploaded_file($_FILES['file']['tmp_name'])) {
    @move_uploaded_file($_FILES['file']['tmp_name'], __DIR__.'/upload/'.$_FILES['file']['name']);
} else {
    @rename($_FILES['file']['tmp_name'], __DIR__.'/upload/'.$_FILES['file']['name']);
}
}