I'm not sure about how to hurry it along. It may not want to bloom this year, as it the climate where it came from may be sufficiently different.
The resulting radish can be saved and planted inside, or saved for re-planting next year. As long as the radish doesn't dry out (perhaps by storing in a plastic baggie), it should remain viable for an extended time.
I collected seeds from a green daikon purchased from a local grocer using this method. I didn't have room to store the entire radish, so I only stored the top four inches. I planted the radish top a few weeks later when my area had warmed up.
It grew leaves and
eventually flowers, producing enough seeds for me to try growing it the more standard way. The tuber won't grow, but
new roots will grow from the exposed vascular tissue.