The US taxes you on US stuff and foreign stuff.
The UK taxes you on stuff in the UK, or income brought to the UK from outside.
So either way you're paying US taxes. One way you're paying UK taxes on only what you bring to the UK for use.
This seems a common belief but isn't quite true, so I'll try to clarify a little based on what I have been told. Bear in mind I'm also not an expert, this is just what I've been able to figure out as I understand it.
The UK does tax you on your worldwide income if you are a UK resident, just as the US does. The difference is that the US will always tax you on your worldwide income, wherever you live, as long as you are a US citizen whereas the UK taxes you worldwide only while you are resident there.
So for example, I'm a UK citizen and my wife is a US / UK dual national. While we live here in the UK she is subject to both UK and US tax on her worldwide income, but if we were to move to the US she would no longer be taxed by the UK on her worldwide income. She would only be liable to UK tax on any UK sourced income, for example property or non tax sheltered investment accounts held in the UK.
What LordSquidworth may be referring to, there is an exemption for foreign sourced income available to imigrants to the UK, but this is only partial. As I understand it you can claim a tax exemption by filing on a remittance basis, meaning you only owe UK tax on foreign income if you actually bring the money into the UK. But you can only claim this exemption if you meet certain criteria - I think if your overseas income is less than £2,000, and you have been in the UK for less than 7 years.
One option that may work for you is to make large contributions to your UK employers pension - the maximum is £40,000 but you can also use the last couple of years contributions if you haven't already. This will get you some pretty useful savings on UK tax. The government has recently removed the requirement to buy an annuity, but the remaining big disadvantage is that, unlike a 401k you cannot make withdrawals before pension age. This is currently 55 so if you are say, 45 or 50 and planning to retire soon it could work very well. If you're under 40 then chances are the age will have gone up to 60 or 65 before you get there. I haven't found a way to transfer a pension to the USA and treat it as a 401k. Note that a pension provided by your employer is exempt from PFIC that CPA Cat talked about, but a personal pension (SIPP or Stakeholder) that you take out yourself may or may not be - even the professionals seem to disagree about this.
Bottom line is that there is a lot of complexity here and after a good bit of digging I've struggled to find any clear answers!
I can recommend greenbacktaxservices.com who prepare both UK and US tax returns for my wife; they can advise on tax from both sides of the pond and are relatively affordable for an expat service. They have given good and prompt advice on questions we've asked them like "what will our liability to UK tax be if we sell a rental property in the US?" but we haven't asked them these specific questions yet as we have the workaround of just making sure I own any UK based investments.