How long does Steam take to send gifts to friends?

If payment went through successfully, the game should arrive instantly as it is an automated process.

I was always able to send the gift right after the winner accepted my friend request.

Also, pay attention to price differences between regions.

Gifts don't have market holds, so it's immediate. On maybe two or thre occasions I've had to restart Steam to actually get the gift prompt, so it may be necessary to actually inform the recipient that they're getting a gift, if they don't appear to have noticed such on their own. Also, once a gift has actually been sent through Steam, your recipient actually has thirty days to claim it before Steam reverses the gift. As such, there's little risk of losing out on any sale pricing, so long as the gift is sent out before the sale ends.

If you're pressured on time before a sale ends, you can remind people who enter to send friend requests your way as they enter the giveaway, so that you can easily and reliably add the winner to Steam in time (and then just mass-ignore all other friend requests received). Given the unreliability of public entrants as far as reading descriptions, any sales which are that tight on time should perhaps be directed towards trusted groups or the like, instead.

As far as sales that aren't as tight on time, they're generally a bit more managable. If you do a flash public giveaway for a game that has a one week sale period and the sale is still fairly close to its start time, it's unlikely (given their previous behaviors) that staff won't just round up to a week, so long as you can give clear indication that you tried to give the game immediately and noted the time pressures in the giveaway description to begin with.

As a more critical reminder, do note (if you're not already aware) Steam's region restrictions on gifting, based on relative currency values. To avoid issues, you can load the intended gift up on SteamDB, which offers automatic currency comparisons between regions, and then just restrict out any regions with a currency value 10% or more higher in value than your purchasing currency. [In other words, if a game is $40 USD and also 40 Euro, and the adjusted value for Euro at that time puts it at over $44 USD, then you wouldn't be able to gift from the US to Europe.]

Thanks for your comments people, I will finally do a short giveaway with region restriction, I hope I have no problems and the winner responds quickly. Fingers crossed! :P