It typically takes about a week to 10 days for your contract and points to be loaded into Disney's system. Then it can take another couple of weeks to get the member number in the mail. You used to be able to call member administration for your member number, but apparently, there aren't very many CMs willing to give that out over the phone anymore.
We purchased a resale contract and it took 57 days from initial offer to having access to the points on the website. All went smoothly for us and there were no unusual delays.
But when you actually stay in a DVC resort and when you "can use" your DVC points might be something else. Just because you have the points doesn't mean there will be availability when you want to book. Especially if you want to book something during the DVC busier times (like early October through mid-January).