Just have 2 of your game at the root folder then. One for the current one for the updated. When you upload a new version you will upload it to whatever one is not live - Once that version is ready you go to your website editor and set the newly updated to the game the website is playing. Next time a user refreshes or logs in they should get the new version but no be able to get there mid upload anymore.
Keep the old version - until you have the next update - this is where you will update the new patch. You will just switch back and forth. Additionally, this will give you -as the dev - the ability to go back to the last patch if you put out a game breaking bug.