I moved over and had a small amount of furniture, dishes, clothing, etc, and if I were to do it again, I wouldn't bring anything except sentimental items that couldn't be replaced. Most things will be cheaper to sell and replace than move. You'll also be waiting months for the items to be delivered if you do a shipment by sea. My items were packed up in the US in late May and I didn't receive them until late October. I'd recommend doing a small shipment with items you can't replace, but leave the furniture behind.
Don't bring any electrical items that operate on a motor, like a blender or vacuum, because they won't work. Things like laptops and TVs will work, though.
As everyone else has mentioned, homes are much smaller in the UK and you may struggle to fit an entire bedroom set into a UK home. My master bedroom, which is considered "spacious" in the UK, is only about 8 feet by 11 feet, and I have an Ikea super king in it, so there's absolutely no room for anything but the bed and a dresser. Non-master bedrooms in UK homes are much smaller and often have room for nothing more than a double bed. Some homes we looked at wouldn't even fit our bed in the master bedroom.