At the time when these events took place (late 18 century) divorce was extremely rare, in fact only very wealthy people could even contemplate it and you needed a special Act of Parliament to get a divorce. It was also considered a disgrace to your family and socially, a person who got divorced would be an outcast. Even for men it was very unlikely; for women it was practically impossible, especially as a husband's adultery wasn't considered grounds for divorce anyway (though a wife's adultery could be). Also, any property a wife had belonged to her husband once they were married (she could get a legal financial settlement, but essentially her property was his) and children were also the property of the father, so in any separation it was highly likely that a wife would lose her children completely.
There was no real divorce law in England until 1858, but even then the law remained heavily weighted in favour of the husband. In Georgiana's time, it would have been quite impossible for her to divorce the Duke of Devonshire, let alone marry anyone else.