I have saved turnip seed a few times. I sow late, July is good, and overwinter with cloche and fleece or a double layer of fleece. Perhaps I have been lucky with my varieties, but they have always come through winter ok with frost protection. Including last winter, which was cold. I don't know yet how well this year's will be doing as this winter is even colder.
Turnips still need a long time to produce seeds the following year. It is not a quick process.
I'd be very interested to hear whether you will be successful in saving seeds in one year.