Each bank or servicer is different. Some banks answer the offer with an approval or disapproval within 3 weeks. Some banks are so confused internally that after 5-6 weeks they still don't have an answer. And, like Don, I had one that dragged on for 14 months and then when we were finally going to get approval (verbally told that), the HOA foreclosed and messed everything up.
As a seller, you should expect at least 3 weeks longer than a normal sale. A normal sale takes about 2 weeks for a cash offer, 5 weeks for conventional and 7 for FHA/VA, typically. The approval will tell you that the normal process is starting and you can ask your Realtor how much time after the Amended Execution date (bank approval) you have to close. You will have to leave by closing.
The buyer can make a deal with you to allow some variation from the move-out and often this is a couple of free days (no cost to you).
It is sensible to prepare (find a place), but to minimize your cost, you should not actually begin the process to move until bank approval is granted.
If the bank counter-offers the buyer (disapproves), then the clock is still running until the buyer raises his offer to meet the bank's counter. Typically, the minimum 3 weeks is when the counter arrives and another week or so is needed to resolve the counter and get it approved. Sometimes buyers back out before approval or refuse to raise their offer to meet the counter.
Finally, banks frequently offer cash for keys. That is, if you move out on time voluntarily, they give you cash. It usually is enough for security deposit on a rental and ranges from a few hundred to a few thousand. I've never heard of getting more than $5k.