Actually, you do have to have your ILR, but I don't think you have to have it for one full year before applying for citizenship. Peedal on this site just got her ILR this past week and is planning on applying for citizenship in May.
I think the rule is that when you apply, you have to be free from immigration control (which is what the ILR means) and you have to have lived here for three years. (The three years starts counting down the moment you arrive with a settlement visa.) For most people this means it'll be one year after they get their ILR, but there are some cases where that isn't necessarily the way it work as they'll have been here on a different kind of visa before they got married, etc.
As for how long the process takes, once you send your application in it really depends. I know there are people on this forum who got approved within a month or two. I did my application last May, didn't hear back from them until late October, and didn't have my ceremony until January. And it's taken until last week for me to get my UK passport. But I think my case was a bit more drawn out than usual for some reason. In theory, it shouldn't take more than 5-6 months.