Is it permissible for a woman who has to make ghusl (full ablution) to breastfeed her baby, to do the cleaning and to do the washing-up?
A woman who has to make ghusl can breastfeed her baby, do the cleaning and do the washing-up before making ghusl when it is necessary. Not having ghusl does not prevent her from doing things like that.
However, our scholars have warned us that we must not delay making ghusl until the time of the current obligatory prayer is over. Not to make ghusl although the time of a prayer elapses means to miss a prayer. It is not permissible to miss a prayer without an excuse.
Therefore, a woman who has to make ghusl can do necessary chores, do the cleaning, even cook but she must make ghusl as soon as possible; she must not delay ghusl until the time of the current obligatory prayer is over. The Messenger of Allah reminded us that the ground that a junub person (a person who has to make ghusl) steps on would curse him and warned us not to delay ghusl in the hadith that he warned those who delayed ghusl without an excuse.
However, a person who needs ghusl is not najis (impure, filthy). He is dirty spiritually but he is not regarded as najis. When Abu Hurayra (may Allah be pleased with him) asked the Messenger of Allah a question about the issue, he received the following answer: Subhanallah (Allah is free from imperfection)! Is it ever possible for a Muslim to be najis?
Thus, a junub person is dirty spiritually but that dirtiness does not mean he is najis, which involves being disgusted.