There are mixed claims in the question, _"lack of sunshine"_ does not mean _"above average rainfall"_ nor _"more rainy days"_. It means more days with significant cloud coverage (some of which may be rainy).
As you can see on below map, insolation (average sunshine energy received) of UK is significantly lower than rest of Europe. !enter image description here
There are two factors determining insolation, weather and latitude. Since insolation of UK is much lower than of other European countries at same latitude, and similar to countries much farther north, only factor responsible for that difference can be bad (cloudy) weather.