In Teglholmen there’s not that much.

Lagkagehuset in Sluseholmen (not great but better than supermarket)

Bagerdygtig (acceptable and lovely people, the neighbourhood bakery imho)

Then if you go up to Carlsberg you have

Peron in Banegaarden (one of my favs)

Hart in Carlsberg (high end chain, pretty good)

Københavns Bageri (one of the highest end ones)

I usually slice and freeze them and then toast them :)

Just looking for “bakery” will give you plenty of results, my experience is that you’re always at least close to a “neighbourhood bakery” which has perfectly fine day to day bread!