Are there restrictions on taking physical gold out of the US if I emigrate? Can I just pack it in my luggage and get on a plane to UK?
The US Border Patrol website says this: " Report Traveling with $10,000 or more: There is no limit to how much currency you may take in or out of the United States; however, U.S. federal law requires you to report your total currency of $10,000 or more. Currency includes all forms of monetary instruments. Travelers who fail to truthfully report all of their currency risk their currency being seized, and may face criminal charges. "
The UK Border Agency website has a less simple answer involving the EU and Non-EU countries. The limit for declaration is over 10,000 euros. The declaration form link and more details can be found here: https://www.gov.uk/bringing-cash-into-uk
The airlines say not to put valuables in checked luggage. But both passengers and carry-on luggage are searched for metal. Obviously, the gold owner does not want the security gate personnel to be juggling their gold.
Also, if the gold were in carry-on baggage then every time someone was pulling something out of a bag, the gold owner would want to stand-up and watch their own bag. So the gold should be carried in a sling on the gold owner's body and under a coat.
A kilogram of gold is worth about $45,000. Suppose an investigator asked for an origin-of-funds ? And so paperwork on the gold is recommended.
If the gold is shipped as cargo then it can be insured. Then both the shipping company and the insurance company could have requirements.