Hi there, ladybug.
Your children should use the US passport when entering and leaving the US, and the UK passport when entering and leaving the UK or other European Union countries. This is fairly important, in terms of the US, because it is actually an offense for you to use the "wrong", i.e. non-US, passport for children with US citizenship. Here in the UK/EU, using the US passport can create confusion in terms of the entitlement of the children to be in the country. If they are citizens of both countries, they will presumably have no visa status for either (being citizens, they don't need it). So you should use the US passport for US purposes, and the UK passport for UK purposes.
The authorities should not have a problem with it, because dual citizenship by birth has been recognised by the US for many years (in fact, I think it has always been recognised).
In the early 1990s the US then also started allowing its citizens to retain its citizenship when they naturalised, i.e. applied for foreign citizenship, under most circumstances.
I am a dual citizen (born in the US, and UK citizen by naturalisation) and haven't had any problems over the last four years since I became a British citizen.
Note that there is a special vignette you can get for the children's US passports, called a "certificate of entitlement to the right of abode". Ask your UK immigration lawyer about it or check it out on the IND website. It is a sticker placed in the foreign (=US) passport which alerts UK immigration to the holder having UK citizenship or a similar status.
I haven't done this yet, but I was told by my immigration lawyer here that I should get one some day, for example, just in case my UK passport gets lost or stolen.
