The mother is 93 with dementia and UTI. She lives alone and should not be. What is the first step to take for the family to intervene??

Call in social services. If your mother has dementia and a UTI eating out of date food she is no longer safe to live on her own.

it will only unfortunately get worse and she will need round the clock care.

First step is to request a welfare check. The family can contact APS and take advice on how best to go about it; bear in mind that unless circumstances are quite extreme APS won't even attempt to enter the house without the lady's consent. But it's a start, leading to professional eyes on the situation and facts on record.

How do they know she has a UTI?

If no one is moms durable or medical POA, then there really is nothing that they can do, unless they want to report her to APS,(or apply for guardianship which is costly and timely) and have them come and check out the situation and take things from there.
If no one has her POA's then they will just have to wait for something bad to happen, and she's taken to the ER or hospital. It will be then that they can let the social worker know that she cannot return home.
If she has a UTI, she definitely needs to be on an antibiotic, as if left untreated can lead to sepsis or septic shock, or even death. Perhaps one of her children can try contacting her doctor or talk her into going to the hospital to be treated. Then they can get the ball rolling to get their mom the help she needs.

How does she have a choice in the matter? One of her children needs to take action and place her wherever she needs to be.