In people who inject insulin there can sometimes be a need to increase insulin levels during the cold winter months and decrease it when the weather warms up, but I'm unsure if cold weather effects other diabetics the same way.
As far as I'm aware the temperature of the water when washing your hands will have no influence on bg meter readings, as long as the hands are clean and free from food debris the result should be the same, unless I'm mistaken...........
Anyway, hope your gp finds the cause of your finger discomfort.