Mark Mayo's answer is the best answer to this question; it is an appeal to common sense. I simply wanted to add some legal context for the UK. There are intersecting regulations for the EEA as a whole which are roughly similar.

The 'generic' answer is that gun permits are generally issued by individual states, like Kentucky or Arizona. Individual states do not have the constitutional authority to engage with foreign governments and this limits their jurisdiction. Notable exceptions are...

where the 'Doctrine of Comity' is engaged (e.g., driving permits, marriages); and

where the Hague Convention is engaged (e.g., divorces, child support)

The UK (along with the rest of the EEA) participates in both of these subject to national policies, like that for driving permits. Gun permits are not taken up in the Hague Convention. Using comity, you might find a country willing to recognize a gun permit issued by a US state, but for comity to work, that state would have to offer reciprocity and states cannot do that for the above mentioned reason.

So much for international law, on to specifics for the UK...

The relevant publication is Bringing Goods into the UK and it's targeted to Brits and permanent residents but has valuable information for visitors as well. The enforcement arm is HM Revenue and Customs and the controlling legislative reference is the Commissioners for Revenue and Customs Act 2005.

The gist of this is that you need to declare it well beforehand and present a UK type of permit for it. Assuming you get far enough along in your arrival to select a declaration channel, you would select the RED channel. I have used that channel a number of times and the HMRC staff has always been friendly and accommodating. So it's possible in theory, but the relevant permits require a permanent UK resident to sponsor your fire arms application. Gun permits in the UK are issued by the police and the visitor must prove they are attending a 'shooting event'.

That addresses the legal side of things...

The remainder of this answer assumes that you are a naive traveller who is carrying prohibited fire arms into the UK with the assumption that it's ok because you are an American...

If you are carrying a fire arm into the UK, your carrier is obliged to report it to the Border Force in order to comply with a Statutory Instrument they are subject to. They will do this before the flight arrives.

At the border, your fire arm will be placed in their custody and logged, cataloged, recorded, wrapped in a plastic bag and nicely stored for you (in a permanent way)...

Source: The Independent, fair use

So you can show them your gun license and explain the Doctrine of Comity to them and after they stop laughing, they will get down to business...

They will use various provisions in the Criminal Justice and Immigration Act 2008 to write you up and send you back to the US. They will then use various provisions in the Immigration Act 1971 to make sure you will never be back. After that they will use the Five Eyes Treaty to let their friends and allies know about it. They will also tell the DHS. You will not have to worry about questions like this in the future because most of the civilized world will put a stop flag on your biometrics.

See also: https://travel.stackexchange.com/questions/tagged/firearms

Also, be sure to read the Chief Inspector's report.

Source: Daily Mail, fair use.

Adding parenthetically... This answer comes from an American (who holds two other nationalities) who has spent long hours of pro bono time at Heathrow and Gatwick providing legal assistance to Americans (and others) who have been placed in detention and awaiting their paperwork prior to removal. Observations have been derived from first hand experience.