Hi Kai,
If you are a US citizen you have to file a 1040 every year. You get an exemption of $80k for foreign income. Even then, because of the tax UK/US treaty there are ways to avoid double taxation.

If you have a Green card and spend even one day in the US during the tax year you are liable to US tax. I think you have to actively renounce the Green Card to not be resident for tax purposes. However, I also believe that if the IRS thinks you are doing this to avoid taxes they might not let you do this and may still require you to file a 1040. In the UK you and your wife will be required to pay taxes per your UK status, i.e. as UK residents.

I think the complications of dual citizenship and taxation is one situation where a qualified tax specialist is really useful. There are just too many potential pitfalls and a few hundred dollars spent on getting an expert to file your taxes is worth the saving in worry and the avoidance of dual taxation.
See you again