A few notes about how long it takes to get your flexible PCB. Your experience may vary.
OSHPark aggregates your design onto a larger PCB for fabrication, then breaks your portion out and mails it to you. The steps are (OSHPark usually emails you a notification for each) 1. assigned to board 2. board sent to fab 3. board received from fab 4. board shipped to you.
My experience with rigid PCB is that the whole process takes about a week to ten days. The longest step is at the fab. Often, my boards (which are tiny) are assigned to a board and sent to fab within one day.
With flexible PCB, the whole process takes about 3 weeks. The longest step is "assigning to board". My first flexible PCB took two weeks to be sent to fab. I suppose that the demand for flex PCB is so much less than for rigid, that it takes longer before a board fills up with aggregated designs. Also, I suppose that most flexible PCB's are small (because they are more expensive, so often only the part of the circuit that needs to be flexible is submitted and the rest of the circuit is implemented on rigid PCB.) That also would mean it takes longer for a larger board to fill up so it can be sent to fab.