My mother was born (1944) in the USA to a British Father and an American Mother. She wants to know what she needs to do to become a UK resident.
She was born in wedlock, her father was born in the UK to UK citizens.
For questions like this, I would normally steer you to: https://www.gov.uk/check-british-citizen. For people born before 1983, things are messy. The two issues are if her parents were married and if her father's citizenship could be transmitted to his children. For example, today, citizens by descent do not automatically pass on citizenship to their children. Your best bet is to talk to an immigration lawyer familiar with UK law.
As your grandfather was born in the UK to UK citizens (i.e., your great grand parents) and was in wedlock with your grandmother, she is eligible to register as a citizen. Given her circumstances her citizenship will not pass down to her children. She cannot become a resident until she registers as a citizen. I cannot find any guidance as to whether she can register from within the UK.
Based on the information provided after the edit, that she was born in wedlock and her father was a British citizen by birth in the UK, she is almost certainly currently a British citizen. She can apply for a British passport for purposes of moving to the UK.