I have uncovered a few more details about Alice and the man she was living with in 1899 which describes her living situation prior to the suicide attempt.
According to the Islington Gazette article (published Oct 16 1899), Alice had a severe drinking problem. She had been living with the "somewhat older" man (described in later articles as being of "foreign" appearance) for four months but the two were not married. He claims that he wanted to marry Alice but was not willing to do so until she was cured. When the man mentioned these details to the officer, Alice began to sob and accused the man of being a "brute" who "kept throwing the past in her face". He told the officer that he wanted Alice to get help for her alcoholism even if it meant that the court would have to send her away for a few months.
The officer took Alice to chemist where she was treated for the poisoning attempt, followed by the workhouse infirmary.
The officer was unsure what to do with the child, but agreed to leave her in the man's care because neither of them knew who the biological father was. The man seemed adamant about being willing to care for the child while Alice recovered. This explains why there is no record of a child entering the workhouse with Alice. Based on the officer claiming that the man was not the child's father, I am led to believe that the child did not resemble him.
Approximately one week later (article published Oct 22 1899 in Lloyds Weekly Newspaper), when Alice's case was heard in court, the man claimed he was unable to care for the child and cited this as a reason why Alice should be released on bail. It is then alleged that the man was selling Alice's belonging. Based on this, it seems safe to assume that they had ended the relationship (which would explain why he would no longer be willing to care for her child).
After this point, there are contradictory accounts of what happened to the child. One article claims that there was no choice but to send the child to the workhouse, where as another claims that the child accompanied Alice to the benevolent home.