I'm moving to the United Kingdom. What will happened to my apartment lease here in the United States? Will that debt follow me?
It sounds like you intend to stop paying rent and vacate an apartment before your lease expires. Technically I don't believe this is a debt, as you have not borrowed any money. Rather, this is a breach of contract. You should probably seek legal advice on what you may be liable for and how likely that judgement is to be enforced overseas -- not financial advice.
Regardless, you might also find it preferable to sublease the apartment or come to an arrangement with the landlord to modify or terminate the lease gracefully. This is especially the case if you are a US citizen or if you intend to return to the US in the future.
Alternatively, you may be able to pay a fee to get out of your lease early. It is typically a lot less painful by paying the fee then dealing with any consequences.
It depends on who you are renting from really. If you are one of these corporate run complexes expected a negative report on your credit. If you are renting from just a regular dude, I would talk to him and ask him if he would let you out of the contract.
Will the debt follow you? What the landlord might do will depend on how likely it is they will be able to collect on a judgement, the cost of legal time in a given locale, and monthly rent. State laws vary but in general it is unlikely a judge would award more than 2 months rent in damages for a broken lease contract. A court filing and attorney fees to get a judgement in a large east coast city will run roughly $1000. If you have no assets or income in the US collecting on a judgement would be infeasible. You do the math and weigh the risks...