None of the veg that I grow indoors need pollinating, that would be hard work and I'm in favour of an easy life.
Overwintering plants that I have successfully grown in the conservatory (so far) are tomatoes, French beans (obviously), carrots, spring onions and strawberries. The strawberries were new this year and they started ripening at the beginning of April and are just coming to an end now. They are grown hydroponically though and the water is heated with a small aquarium heater and kept at a constant 20c regardless of the air temperature.
I have the next lot of strawberries grown in soil starting to ripen now. Rather amazingly there won't be any gap between one finishing and the next lot starting, I wish I could claim that it was good timing on my part.
The overwinter tomatoes were also grown hydroponically which survived and ripened fruit at temperatures close to freezing, which I'm sure was only because their roots were kept at 20c.
My "normal" overwinter tomatoes get the luxury of a windowsill in a heated room.