Since their birth (10 years) I have been saving money into a tax free account for my grandchildren in a UK bank. I now wish to transfer the lump sum to them in USA. They have UK and US citizenship as an American mother and British father. Can the tax-free status be maintained or are there taxation implications in the USA?
It's difficult (impossible?) to properly answer this question without knowing more about what kind of account it is. Based on "tax free" I'd have guessed a JISA (Junior ISA) or a Child Trust Fund but I'm not sure that can fit with the "two signatories" one of whom is a grandparent. Or is it not actually tax free but just a regular savings account paying interest gross?
In any case, I fear you may be sadly mistaken about its tax free status for the US citizen part-owner. US citizens (including children) are subject to "worldwide taxation" and my understanding is that the IRS does not recognize things like the JISA tax-free status. (There's an interesting read at this "What Americans in the UK need to know about taxation for children" page which gives some idea of the complexities involved, but which may also offer some solutions e.g a "section 529 college plan" is mentioned).