EasyJet are fine about this at Gatwick, we've done the late night bag drop for some years now, as we tend to be on a 6am flight, so drop the bag(s) off any time after 8pm the night before. Means that we can go straight to Departures first thing in the morning, without having to join the baggage queue.
You do get charged if you are overweight (your baggage, that is, not the passenger) and you can only take one bag on board (so handbag goes in the hand luggage) but I know of no hidden charges, they are all clearly laid out.
How long it takes to drop your bag depends entirely on how many people are ahead of you in the queue - we usually go over at 8pm and there are only two or three people there ahead of us. You need to take your flight details and your passports with you - the process is exactly the same as if you were checking your bags in on departure, they give you your boarding cards and the luggage receipt when you hand over your luggage.