Tomorrow: Main risk is for Kent and East Anglia and continuing up the east coast as weather pushes up from France. Possibly some heavy snow arriving by morning in Kent and lunchtime or so in EA, Yorkshire by late evening. Also to confuse things there may be some heavy snow showers in the NE for a time during the day too.
Other eastern areas (perhaps as far west as Hampshire) will also be affected by this but to a lesser extent. Other areas, the possibility of some flurries.
Monday: The east coast will be most prone to snow showers but they could pop up in central England too. Perhaps some sleet or rain mixed in, particularly in the south.
Through the rest of the week a very similar picture. Perhaps the south may be a bit too warm for snow from Monday/Tuesday on. Keeping an eye on possibilities for Wednesday and Friday, but at the moment it is too uncertain to speculate. Impossible to say when the cold will leave us, at the moment it looks like hanging on in the north for this week at least.