Hewitt example

A  U.S/UK dual Citizen resident in the UK, receives payments from a U.S. contributory defined benefit plan. Under the new treaty, they will be subject to UK tax (on account of UK residence) but because of the saving clause, they also will be subject to U.S. Tax (on account of U.S. citizenship). This may be subject to adjustment to avoid or reduce double taxation.

Can anyone tell me which country gets the tax and which the tax credit? The common sense answer is the tax goes to the US
and a credit is applied for in the UK. Does the tax law say soemthing different?

This is way too complex for this forum...I'd suggest you take a look at the other posts in other parts of the forum.

A good US/UK dual qualified adviser can provide this advice for a couple of thousand pounds or so.  Alternatively you could ask HMRC as they will be delighted to help out so you don't pay more tax than necessary.