Your child may apply as a minor at the same time for Austrian citizenship as you do it (although not on their own, but by extension (durch Ersteckung) in conjunction with your naturalisation. If your child has not taken up their British citizenship, to which they have a claim by descent (durch Abstammung) they would no longer have a claim to it once you naturalise and renounce your British citizenship. This of course would be of relevance to hitherto unborn children (I have children born after naturalisation who are Austrians by birth/descent (durch Abstammung – see Article 7 StbG – checked on 05.01.2019), who would have also qualified for Austrian citizenship in the unfortunate event that I had died after becoming an Austrian citizen but prior to their birth. This last point might sound trivial, but made a difference to us, given that my wife was stateless at the time of the birth of our twins, only 8 weeks after my naturalisation, by provided that I had been an Austrian citizen at the time I died would have received Austrian citizenship.
There is no automatic mandatory naturalisation of your children if you choose to naturalise “Zwangseinbürgerung“, although some other citizenships demand that minors of parents changing citizenship also take the new citizenship (this is the case for Russian citizens). When my son’s naturalisation was done with mine, my wife and I had to provide a signed declaration (non-notarised) that our son had not taken up Russian citizenship in the mean time (even though this of course was not possible!)