re: "I have completed that process but didn’t know if I should upload the completed document or wait for some sort of payment to come through the system."
You should provide the work product immediately.
All work product generated by you while logging time belongs to the client who is paying. So in that sense, the files you created are not yours. They already belong to the client.
Keep in mind that the client provided his credit card information to Upwork already, and your logged time will already be billed to the client, and the money paid to you. The client does not need to do anything else for that to happen. You don't need to do anything else for that to happen.
So, yes, the correct thing for a professional freelancer to do while using an hourly contract is to provide everything to the client immediately.
Personally, I provide client access to an online space where I actually do the work, and my clients have complete, real-time access to all the files I generate, all the time.