It took me a long time to get used to it, too, and I'm from Chicago originally.
It makes me really laugh now to say this, but the thing that helps me the most is... drinking tea.
It's hilarious because when I first moved across, I thought tea was vile and drinking it actually made me feel queasy, and now, four+ years later, I'm like "hey, boil the kettle, will you?" I have been assimilated.
Never heard of strapping on your hot water bottle, but my MIL in Belfast has sent them to bed with us before to keep us warm. Seems everyone sells beanbag style ones now - might be nice to have on the sofa while you're home.
For going outside, I have always found your first line of defense should be waterproof stuff. I live pretty far north, but I'm near the coast and it doesn't snow a lot, just rains and rains and rains for months. The best coat I have for this is one with a waterproof windbreaking shell and a zip-in fleece lining. It's warmer than any of my other heavier duty ones I bought during arctic Chicago winters because it keeps the rain out. It's nice that it has a hood, too, so I don't always feel like I need an umbrella when it's just that not-really-raining-but-definitely-drizzling-all-day kind of weather.
Also, invest in some good waterproof shoes. I still lament the lack of cheap Payless shoes in my life, but it really is worth your money to get good leather ones here, because they will keep your toes dry. The less soggy you are, the better you will feel, not just when you're out, but when you get home, too, and don't have to put your socks on the radiator to dry out.
I've always found it helps me to own a ridiculously cheerful umbrella, too. My husband wouldn't be caught dead with one (he likes black, plain black) but I need all the motivation I can get to drag my butt outside for some fresh air on a rainy day.