I am going to guess that in most people's cases, upload speed is the issue. There are a lot of asymmetric Internet connections in the world and on those upload tends to be a lot less than download. Also you are going to be uploading an uncompressed WAV to them and not just an uncompressed file, but a high bit size one at that. So you have to toss a bit of math at how long it will take: A 44.1kHz, 32-bit uncompressed stereo audio file is 345KB/sec or about 20.2MB/min. Well, have a look at your Internet upload speed. In most cases, it is probably in the realm of 1-10mbps. However note that is bits, not bytes per second and also it gets more confusing because computers abuse the metric prefixes like mega sometimes using it to mean the proper 10^6, and sometimes meaning 2^20.
Long and the short of it is that if you have a 10mbps upload you will see an effective rate of around 1MB/sec, give or take (depends on network load, kind of connection and a ton of other things). At 1mbps, it is about 10 seconds per MB. So if you have a 3 minute song, and a 2mbps connection you are looking at around 5 minutes to upload to file to them, before they start processing assuming everything is working well.
So just keep that kind of thing in mind when you are looking at times. Make sure to account for the transfer time it is going to take, unless you have a really fast connection.
Also note that if you are on WiFi, that can slow things WAAAAY down. A full discussion of WiFi speeds would be a massive post but suffice to say it is always a lot slower than the data rate it shows because of overhead, and on older standards is often slower than modern Internet connections.