You'll have to make sure that the uploaded data gets written to one single central storage. You can modify the UploadHandler.ashx and change the code that writes the data so that it writes to a single central storage.
Then, it's no problem that there are multiple webservers. Each webserver that receives a chunk of data will write it to the same storage.
I think that's the only way to get this to work in a cloud/multi webserver enviroment.
If you want to make sure that there are no conflicts between multiple users uploading the same file, you can dynamically create subdirectories for each individual user. For example based on username or IP address. So that each user uploads it data to a different
directory. You will also have to modify the UploadHandler.ashx to support that scenario.