Auld Yin is right. Cheaper things like Ikea is probably not wothwhile shipping (although, if you already need a container and there is still some space: fill the space with it, as you already paid for the container it's cheaper to bring it than to buy new).
And certainly if you are having (younger) children, I should ship their things! Moving to a new country is a big thing, and their favorite toys/books/games/decoration will certainly help!
At first, we considered a 40ft container, but after going through our stuff we decided to sell some things and go for a 20ft container. Don't regret that! Although I do regret that we also sold our dining set, because that quality costs a lot to buy new here! And even than it is a different style that I don't like as much as the one I had.
You can always take a look at the Fidi network of international movers. But I have no experience with UK movers. Maybe someone else here?