If you are not passing through London en route to another country where firearms are more freely permitted, your firearm is subject to confiscation and you'll likely be arrested. Flying with a firearm in checked baggage is not so much a problem when it comes to domestic flights. Go to the airport with your shotgun unloaded and in a locked hard-sided case to which only you have a key (think, pad lock). Depending on the airline, your ammunition may or may not have to be in a separate hard-sided, locked case. Go to the airline counter to check in and tell them that you have an unloaded firearm to declare.
I know that hunting is still popular in England and in other parts of Europe, so it is probably a possibility to take your shotgun to Britain. I haven't taken long guns to Europe, although I have a couple of handguns at our vacation home in France. This was a long process, but it is possible if you are persistent and follow their procedures to the letter.
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