I collected a bag of tame chestnuts when they fell in August. A few weeks later I found most of them had sprouted in the bag so I planted them in potting mix and they are progressing. The best result has come from suspending a nut over a jar with its root in the water. That one now has a green shoot growing upwards. My question is will the seedlings keep growing indoors through the winter so they will have a head start if/when I plant them next spring? If I keep a couple in a pot and bring them indoors for autumn and winter will they skip hibernation and keep growing?
There are a few plants which are called chestnuts which are not of that family: horse chestnuts, water chestnuts, chestnut oaks. If you have one with edible nuts then you have a member of the Castanea (chestnut) family.
This article has a lot of detail. Chestnuts need a dormancy (cold) period before germination so I am surprised you have had any success. Any that do sprout should be potted up and you can grow them on during your winter. Put the pot in the ground in the spring and leave them outdoors over the winter. Protect the sapling from mice,rabbits, deer and wallabies (if you are located Down Under) with chicken wire or topdressing with soil or both.
If you bring a plant that needs a dormancy period indoors in the winter it will not get the right mix of seasonal cues (decreasing day length and lower temperatures). Having tried something similar with bonsai oaks I can verify the contents of this link that says
Firstly, all deciduous and coniferous trees need a period of dormancy which is only triggered by cooler temperatures. Without this dormancy, trees can continue to grow for anything up to 2 years before going dormant whatever the season or temperature; this enforced dormancy can often be fatal.
Chestnuts are nutritious and provide food and shelter to your local ecology but "when grown from seed, the trees do not begin to yield fruit until they are thirty to forty years old." See http://www.pfaf.org/user/Plant.aspx?LatinName=Castanea+species.
Have patience and good luck!