How fast do you need to be going to get an effective slipstream on a bike?
It depends. I don't know of any scientific research to support an actual speed.
The main factors are the speed of the wind that you are riding in, and the size of your shield (i.e. the bunny in front who is giving you the wind break - the bigger the better) and how closely you follow. In still conditions I find that even at about 24km/hr (15mph) I will notice after half an hour or so that I have not worked as hard had I been riding out the front.
If there is a raging head wind then virtually any speed will find you having a noticeable advantage in the slipstream. Just try being out for a while at the same speed and you will feel the difference in your heart rate and muscle output.
The size of your shield also helps. Being in a large bunch effectively increases the size of the shield, though it is not a linear increase, but rather a diminishing one. That is, as you go from say two riders to 5 riders you get a fair increase in advantage , but going to 10 riders only gives you a small increase in advantage over 5 riders. The third factor of distance is not as important I feel, provider you are within one to two feet and are not going at top racing speeds. As you go faster the distance at which you get an advantage increases. Of course the maximum advantage is obtained when you are within millimetres of the wheel in front, which is why track racers ride so close, but then you increase you risk to touching wheels and having a crash.
It is not necessarily how fast you go, but how closely you are following the person cycling in front of you. The closer you follow, the more likely you are to trail in the wind they have helped part for you. The effects of this feel greater the faster you and the other person is traveling. The effectiveness of the slipstream is relative to the speed and distance you follow.
Even runners use the slipstream, and they run at a max of 12mph, but I have enjoyed drafting at considerably slower running speeds. It also depends on the strength of the headwind.
And, as Teddy pointed out, the distance you are from the rider in front is important. This is why runners can draft at relatively slower speeds, although for a skilled cyclist (e.g. one that rides rollers all winter) you can ride just about as close as you could run next to someone.
Furthermore, unless you have a direct headwind, you should ride off to the leeward side. For example, if the wind is coming from the right(left), your right(left) hand / shoulder / arm should be as close to the left(right) butt/thigh of the rider in front as you feel comfortable. Put your head down and enjoy. That is the place where you can hang on when the going gets tough.
But remember a golden rule - whoever's front hub is in the lead has the right of way (e.g. if you are drafting, you must yield).