What is the average waiting time for a paper deposited to an open archive such as HAL?
As commented by Davide Giraudo, the waiting time is typically one or two open days. I think submissions are manually checked by a team in France, so if you submit on Saturday Australian time, you'll have to wait at least until Tuesday. If the team is very  busy, it can take a few more days.

Not asked but maybe worth mentioning: to check whether you are allowed to upload your article to an open archive such as HAL, type your journal name on RoMEO to get something like: