Rough out a bow and then stick it on a long radiator that doesn't get too hot (just warm is fine). You can get it seasoned in about a month, you may get some warping so don't make it too narrow, just loose some of the thickness.
If you make a greenwood bow and shoot it, it will be a bit rubbery, it will then stiffen up as it dries and may well break... that's what happened to all the bows I made as a kid.
If you work on two staves, use the worst one for rapid seasoning and that will give you something to play with while the better one is seasoning more slowly. Also get out and cut some more staves this time of year.
Del