What happened to Rose's mother after the sinking?

She likely survived as she made it into a boat. This is supported by the originally scripted scene on the Carpathia.

Cal was intended to actually find Rose aboard the Carpathia after the sinking. Rose was to tell him to let her mother know that she died on the Titanic, and that he would leave her alone for the rest of her life.

And also by the ending sequence.

Several main characters are not present in the ending sequence, including Cal Hockley, Rose's mother Ruth, Bruce Ismay and Molly Brown. This is because they survived the sinking. Only the characters who died on the Titanic are present in the dream sequence, because it serves to illustrate Rose being reunited with all the people she lost when it sank, including Jack.

What happened to her afterwards is unknown, even according to an amazingly detailed "biography" on fandom.

Most fans speculate that she became a poor and penniless seamstress and lived out her life working in a factory. With is very possible, without the financial security of the arranged marriage between Cal and Rose.

It is, however, also just as possible that she re-married into wealth. However, since Ruth does not make an appearance after her survival, nor is mentioned again, her fate is left unknown.

So speculate away, this is one of the many places to do so.

I think she was exaggerating the degree of their poverty. Even if the family fortune was gone, there may have been relatives who could help. She would not have been a starving seamstress. She just would not be able to live in luxury.

Even if she had really had no relatives who could help, right after the disaster she would have had lots of rich friends eager to be her hostess. She could have lived in continued luxury and comfort for some months, especially if she had been wearing expensive jewelry that she could sell to buy clothes with.

Also, the steamship line paid some compensation to survivors (not sure if it was only those who sued, but I see her suing). And there was some sort of benevolent association fund for the survivors. Stuff like this could have kept her going for at least a year, during which I am sure she did her best to catch a husband (even if she had to settle for the upper-middle class, she would have had security, servants, etc.).

Eventually, if she did not remarry, she could have made a living as a paid companion/chaperone. A lot of nouveau rich American families would hire women with the right manners and connections to help them and their daughters make their way into the upper classes. So I don’t see her ending in dire poverty or in a working-class environment.

There is high likelihood that she survived based on the following logic.

When they introduced Molly Brown to the story, Rose said "she would become known as the 'unsinkable Molly Brown'". This implies that Molly Brown survived the disaster in order to earn that nickname. So we can assume Molly Brown's boat was rescued. But guess who else was on that boat? Rose's mother - and she looked quite comfy with all those blankets when they showed her floating on the water, at least compared to the others.

So, yes, she would have survived.