As things stand, the Duchy of Lancaster is tied to the throne, and the Duchy of Cornwall to the heir apparent. (Both are, obviously, in the gift of the crown, but considering they make up a major share of the royal income, that doesn’t seem likely.)
The abdication of Edward VIII/Duke of Windsor does not really set a precedent, in that the decisions made at that time were very much tied to the particular circumstances at work. And aside from that, there have been no voluntary abdications of the British crown (several forced ones, but I don’t look at Edward II, Richard II, or Henry VI as setting precedents either).
When the reigning monarch’s mother is still alive, his/her mother is formally Queen Name, and informally "the Queen Mother.’ AFAIK there have only been a couple of occasions when the monarch’s grandmother survived, but again formally Queen Name and informally Queen Dowager is the custom. Note that reigning monarch, the queen consort of a reigning king, and any surviving queen consorts of previous queens are all “Her[/His] Majesty,” with limits imposed only by how many monarchs can die off and length of survival of the former consorts. For about a year in 1952-53 the U.K. had three: the Queen (sc. Elizabeth II); the Queen Mother, AKA Queen Elizabeth (nee Bowes-Lyon, widow of George VI); and Queen Mary (grandmother to the Queen, nee Teck, widow of George V).
I suspect the present Queen and Charles would work out ahead of time what might be the proper usage – but also there’s a work ethic drilled into E2R from her childhood that abdication is just not done – the only way she ditches that job is to die. (A disabling stroke or something similar might be justification, but just barely.)
As I’ve noted in the past, though, the death of the present Queen with no other changes preceding it would result in an unprecedented situation with regard to style: For the first time in British history, the reigning monarch’s father will be alive. I suspect the designation will be “H.R.H. Prince Philip, Duke of Edinburgh, Father of the King.” But that remains to be seen.