As far as I'm aware, as long as you are legally staying in the country (semi) permanently and are paying your taxes, you have the same rights as anyone else with regards to childcare. In Germany this should include the right to a subsidised place in a Kindergarten or with other similar care takers like a Tagesmutter (semi-private single person Kindergärten for small groups of children, if you will). As a side note: It should also include monthly payments of Kindergeld by the government to you; make sure you try to get all the different benefits the government and/or your health insurance (!) offer. Depending on [the absence of] your income or how long you're taking care of your kids at home, these can be quite substantial.
Availability of childcare may vary quite a bit depending on your location; some may tell you you should be reserving a Kindergarten spot before your child is even born, though this madness is subsiding recently and it's typically possible to find something reasonable. You should make sure you're timing your applications well; Kindergärten often have a yearly cycle and you may have to wait the better part of a year if you miss it and all spots are taken. In many places Tagesmütter fill the void for such cases.
The best thing to do would be to get in touch with your local municipality, they typically offer specific information about your options, including a list of Kindergärten in your area. There often also exist volunteer organisations that help you navigate the bureaucracy, especially if you may not speak the language. Search around for "Elternhilfe" or similar keywords.
I cannot speak for Austria, but I imagine it's pretty similar.