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The Wikipedia page for the list of shows aired by Pogo
mentions a show called "I'm Old Enough" in the former programming section. From the name of the show, which matches your description (kids doing stuff on their own, which can be explained by them being "old enough"), I can assume this could be what you're looking for, but the show does not have a Wikipedia page (it's a red link in the linked page above), and looking it up on Google barely returned any results about it, so it's probably extremely obscure. Still worth a mention either way.

it would air in between two shows and each segment was about a different kid (from different cultural and racial backgrounds) doing stuff on their own. like for example there was one episode where a little girl is shown going with her dad to a location where people are forming a human pyramid and then she is seen climbing it on her own and going to the top and then she waves back. Another episode shows a mom leaving her son and his little sister alone at home and then the boy and his sister proceed to bake a cake. the whole process is shown like sifting the flour and adding the toppings etc. This show did not have any theme song, each segment lasted a whole of w to 3 minutes. It used to air on POGO channel India around the same time as 'Brum'. It was probably a CBBC original because 'Brum' was too.
Edited by GrandioseIsopteran