On a form with multiple upload fields, quickly clicking the upload buttons causes the files to clear and not upload

The problem
On a webform with multiple upload fields, if you choose files for all the inputs, and then quickly click the upload buttons of each input, only the first clicked file is uploaded, and the others revert to their 'choose file' state.

Steps to reproduce:
1. Clean install D7.43 with webform-7.x-4.x
2. Add a webform with multiple file upload fields
3. log out and go to the webform as an anonymous user
4. Click select file on one of the file inputs, and select a file. Don't press upload.
5. Click select file on another of the file inputs, and select a file. Don't press upload.
6. Click upload next to the first file, then very quickly click upload next to the second file.

Note that you have to press the second upload button very quickly after pressing the first, otherwise you wont replicate the issue. I've tried quite a few times and can reproduce this consistently, but it all depends on you clicking the upload buttons in quick succession.

Expected Result:
Both files should upload, and the 'choose file' button replaced with a link to the uploaded file and a 'remove' button.

Actual Result:
First file is uploaded with a link and remove button, but second file input reverts to the 'choose file' state.

Here's a screen recording of me reproducing what I describe above: (this is on a 7.x-3.x install, but exactly the same thing happens on 7.x-4.x but without the JS console error).
https://www.dropbox.com/s/3dupbsxmbquut2d/webform.mov?dl=0

First reported in another issue thread https://www.drupal.org/node/2675170#comment-10906516 and reproduced by @srutland in https://www.drupal.org/node/2675170#comment-10925615