I am wondering if I can start downloading the SCJD project now even before I received my SCJP certificate?
Well, I think so. I think it is possible.
Moreover, when shall I take the SCJD exam?
You'll have to go for it after the upload. If I make no mistake, some people uploaded the project after taking the exam, and as far as I can remember, there were no problems. But in this case, I don't know how Sun deals with that. But people usually take the exam after uploading the project.