A real week of two halves on the way this week with a very cold start to the week but milder air will try and attack from the southwest later on which should eventually bring temperatures up a little but may also lead to some problems with snow and ice.
Starting with the first part of the week, you could effectively summarise Monday to Wednesday in a similar way - it'll be cold everywhere with hard frosts by night as temperatures perhaps dip below -10c in rural spots. Freezing fog may also be an issue, with icy roads and pavements, along with poor visibility in fog affected regions a hazard to watch out for. By day, temperatures will struggle to get more than a few degrees above freezing at best.
We also have snow on the menu during this period with showers falling as sleet and snow affecting the north and east of the country at times, but in the main where fog doesn't linger there should be a good deal of winter sunshine on offer.
Beyond midweek there is a good deal of uncertainty in terms of timing and the eventual outcome, but milder air is expected to try to make inroads from the southwest during the latter part of the week. As that happens there is the risk of some heavy rain, which could fall as snow as it bumps into the cold air and potentially also cause some issues with icy surfaces. So definitely a period to watch...