I guess I can't really see hygge and Christmas wrapping in the same context! LOL
What I found worked well when we had children at home was to realize that there was a LOT of wrapping to do, it would make a big mess and take a lot of time. What I did not want to do was to have crunch-wrapping on Christmas eve. To avoid that, DH and I would plan an evening for wrapping after children were in bed asleep. We would get out all the gifts, the wrapping paper, ribbon, tags, etc. We made a good team. DH would wrap each gift in the paper I selected for it and put a sticky note on top with the item and recipient. It would take me longer to embellish each gift with ribbon, bows, and tags, but I enjoyed doing it. We would just stay up as long as it would take. It seemed like a marathon. Truly, we should have simplified the gift giving.
These days with having to mail gifts and being retired, we still plan a time for wrapping, but it can be during the day rather than an all-nighter. I need to try to find creative ways to embellish the gifts with something that won't get crushed in the mailing or come loose.
The hygge comes when it's all finally done.
I've never been one to wrap each gift as I purchase/make it. We do not put any gifts under the tree, even family gifts, until Christmas Eve.