Upvote:4
Your mother may possibly be OK.
It's probably going to depend on whether the UK authorities are aware of her previous overstay. If they are, then she will certainly have 'problems'. Whether the problems are enough to prevent her getting a visa is something only the UKBC or someone with more experience than me can answer.
If they are not aware, then she may be OK simply because the Standard Visitor Visa Application now only asks about travel within the last ten years. Your mother can truthfully answer "no" to the question about whether she has visited Britain in that time. There appear to be no questions that would require her to divulge her earlier history. Of course if she was deported, removed, or required to leave Britain because of her overstay then she will have to declare it (but in that case the authorities are aware of her overstay already).
She has two ways to find this out more certainly. One is to apply for a visa, answer all the questions truthfully, and see what happens. I would certainly want to make sure she provided lots of evidence that she has strong ties to her home country, both in case authorities are aware of her previous overstay, and also because as a mother with several grown-up children in the UK she will anyway be under suspicion of wanting to overstay.The other is to consult a UK lawyer (beware of people claiming to be experts in this field who are not British lawyers). This will be more expensive than an application, but runs no risk of making her case worse.
EDIT
The news that your mother is a US citizen changes things a bit. Your mother has the option to visit the UK without a visa. She can just show up without having to fill in the forms or get her biometrics done. This does make the process simpler and cheaper, but it increases the risk somewhat, in that if the UK connects her to her overstay then they may deny her entry at the border, forcing her to get on a plane back to the US. If she were to apply for a visa (which you can do, even if you are eligible to enter without one) it is virtually certain that if she is granted a visa she will be admitted.