I am guessing that your mum is no longer around, so you can't ask her why she and her half sister stopped seeing one another.
However is there anyone around who might know, such as your dad, or her friends, or one of her siblings?
I am guessing there must be a solid reason why that contact stopped, and I think you should think very carefully before contacting her.
Divorces in the 1950s and 60s were "fault" divorces - e.g. adultery, abandonment etc. Custody would often be awarded to one parent and the other might not have access. Father's were as likely to get custody as the mother. Attitudes and times have changed dramatically.
Because someone had to be "guilty", it may be that your Gran's 1st husband never said anything nice about her to their children. Or maybe the children felt scorned by their mother, and that translated itself into bitterness and critism when your mother met her half sister.
If you do decide to go ahead with contact, have you thought about using an intermediary, like a local priest?
Maybe you could find the divorce papers and see what they reveal - the National Archives is a good place to start:
http://www.nationalarchives.gov.uk/records/looking-for-perso...