Brown willow is natural unprocessed willow with bark on that has been dried. Brown willow needs soaking for approximately one day per foot of rod length and therefore needs a little fore planning. Steamed willow has already been processed in water to make the skin chocolate brown/black so may require less soaking.
If you are in a hurry you could try soaking in hot water to shorten the soaking time so you have to be prepared for a little trial and error. You could soak the willow in a soaking bag, in the bath (please be aware that tannins in the willow bark may stain your bath) or a makeshift outdoor pond/pool. Also water butts are handy and some people recommend using large drain pipes capped at one end.
Once soaked, drain the willow and mellow overnight in plastic or tarpaulin (plastic sheeting, hessian or an old blanket/sheet is also acceptable). This is important as it allows the willow to absorb moisture into the core fibers and thus remain supple for longer. Whilst working with the willow, please keep it covered so that it does not dry out. The willow will continue to be workable (if kept under cover) for up to a week before it starts to mould, again this is weather dependent. Over soaking of brown and especially steamed willow will cause the bark to shred and the willow to become greasy to the touch. The willow may also split along the length whilst weaving. Stand the willow up to drain then let it mellow again. External weather conditions such as extreme cold or very warm weather will have a bearing on your soaking times. It may take longer in colder weather and shorter time in warmer weather.