Privately owned airports wouldn't have money tied up in large stockpiles of deicing fluid so this could have a serious impact on airline operations.
I would hazard a guess that privately operated airports will very quickly have run the numbers and discovered that the cost of carrying deicing fluid and snow moving equipment in inventory is minor, compared with the instant loss of revenue during a shutdown of operations.
If I remember correctly, I think LHR alone generates GBP 1 million of operating margin in its retail operations every working day, saying nothing of aircraft landing fees, space rentals, passenger terminal fees and car parking revenue! This is not rocket science!