One clue is the photo session of OTMA during Aleksei's christening. In the summer of 1904, the Big Pair were deemed old enough to wear full court dress and participate in the ceremony. Olga was 8 (nearly 9), Tatiana just a few months over 7. The Little Pair, who were 5 and 3 years old, wore only kokoshniki and their traditional white cambric dresses. By 1910, all of OTMA wore full court dress for their formal photos. Anastasia would have been approximately 9 at that time.
So the children of the tsar were perhaps 6 or 7 years old when they began to wear court dress. I have no idea when other court members' children would have worn full court dress. Except for the tsar's own children, I doubt whether any young children would be expected to participate in formal functions at such a young age. I wouldn't be surprised if OT's court dress appearance at Aleksei's christening was something of an exception, seeing as it was in essence a family ceremony.