I'm not a meteorologist either - just speculating.

I believe it gets so dry in winter because, at least where I am -
1) Winters are generally colder than in the UK. Cold air holds less moisture.
2) Heating systems dry out the air further. I may be completely wrong, but I believe when you heat air, it expands, and the percent humidity at 20°F/-1°C will be less when the same air is heated to 65°F/18°C. The gasses in the air expand, but the minute droplets of liquid contained in it don't. Add the fact that most Americans in cold climates try to limit the amount of fresh air coming into their homes, so less moisture replenishes what may come out of the indoor air.

The general reason for the humidity difference between the UK and US is how cold the winters get. This factor varies widely within the US, and some places in the US would probably be very similar to the UK.

I believe tuning problems happen when the humidity changes. The instrument starts to dry out, the wood shrinks a bit, and you have to re-tune. The opposite happens as well when the humidity increases and the wood expands. Once the humidity in the air and the instrument reaches a sort of consistant level, the tuning is back to normal.