The first week of February will be cold across the North of England for the most part. There will be some prolonged spells of rain or snow at times, though any lying snow won't lie for long. Towards the end of the week it will probably become milder, but just how mild is uncertain.
The second week of February is likely to be cool and wet for the most part, until easterly winds start to try to nudge in during the second half of the week. Whether they succeed or not is uncertain, so it could be either cool and wet or cold and snowy.
The third week is likely to see northerly winds, but rather than a northerly blast as seen at the end of January, some fairly cold and nondescript weather is likely. Confidence is very low.
The final week of the month is likely to remain on the cool side, with further boring conditions. High pressure will be close by, but there is always a good chance of dull, cold weather in winter, though of course it will be nearly Spring by this stage.