Can an iPhone be force started before the 15 minute recharge period?
Unfortunately there doesn't seem to be a way. As far as I know it charges to around 4-5% of battery capacity so that it's sure to be able to perform the whole boot process, even if it's unplugged, so not to screw something up when losing power while starting up the OS.
15-18 minutes seems long to me, though. It's usually 5-7 minutes using the iPhone charger.
Are you charging off a PC USB port?