Not to our knowledge or experience in the UK.
We have been designing and installing reed-beds for 20 years. Not once in that time has any one of our more than 200 clients contacted us to say that their reed-beds had frozen up and were not working. One of our earliest systems was installed just below the tree line on top of the Pennines near Alston in Cumbria. The clients reported that during the first winter they had 90 days of continuous frost. Some ice formed on the top of the beds but they never stopped working.
As the septic tank is underground its contents will be at or above the surrounding ground temperature which is 10-12 degrees Centigrade. In addition, the tank will receive a certain amount of hot water which will help to keep the temperature up. What this means is that doses of effluent applied to the reed-beds are warm and, if the beds are covered in snow, will not freeze. Only in severe freezing conditions and in the absence of snow, which acts as an insulator, is freezing a possibility. We design and install our systems to minimize the chances of this happening.
Small domestic and large commercial reed-beds are used extensively in France, Austria, Germany and Denmark where the winters are more severe than in the UK.