If it is possible for volunteer tomato plants to produce a crop in your location, you may have a good chance. You will have to check with your new neighbors, WildeHilde.
An advantage your plants could have over an outdoor volunteer is benefiting from TLC. However, I don't imagine that you can make up for cold temperature during the hours of darkness by bright, sunny days. It would probably be best to bring the starting containers, indoors.
I have found that the top of my fridge (not in the fridge but, on top) is a place that stays above 70Ã‚Â°F, 24 hours each day. I start most everything up there. Gotta watch them carefully because when the sprout - they go searching for sunlight and stretch!
As soon as they are above the soil, off they go to a south window where they spend a few days before going out to the greenhouse.
If you baby them - they may produce a crop. Mostly, that will depend on the length of your growing season. If you won't have frost until mid-October, many early-maturing varieties should have time to ripen fruit.
We are each other's harvest; we are each other's business; we are each other's magnitude and bond. ~ Gwendolyn Brooks