My family traditions in fruit tree grafting are from East Anglia, England. The traditions date to way before the late 1800s, but that's when my great-grandfather learned, and most of what follows comes through him (and his brother the
Plesher - guy who makes and maintains hedgerows). Note this is traditional oral wisdom, not modern grafting ideas. It's also specific to that climate - which was far more severe than it is today due to it being the end of the last mini-ice age.
First, it was sacrilege to cut a tree if there was a likelihood of frost. Cutting the tree in winter causes
enough risk as the tree is dormant and is slow to heal the wound. Cutting the tree and having a freeze within a couple of days greatly increases the chance of die back or infection entering the wound/cut.
The fruit trees were pruned in late fall or early winter after the barley was in and the kitchen garden put to bed (weather dependent). Several branches from the pruning were kept for grafting and these were placed with their butt in the soil like a cutting in the shade of the tree.
When grafting season came, these sticks were taken from the ground and used for the scions.
When I read modern
books about grafting, they say this idea is bad because the scions will get damaged. Maybe in different climates, they do. All I know is it works for us. The branches freeze during the winter, but not for the first few days (because we don't prune if there is a risk of frost) and somehow acting like a cutting keeps it as living tissue which is more resistant to frost damage than recently cut tissue (think kale that can withstand a frost, but will go mushy when harvested frozen).
To
answer the question more directly, we avoid the freshly cut branches freezing both on the tree and scion until they have a chance to recover.