The show's only legal obligation is to provide the prize as described; you may accept or forfeit depending upon whether or not you think any added costs (transportation to the departure city, taxes, etc.) are worth incurring.
My experience has been that most prize brokers and networks are usually happy to accommodate most reasonable requests, provided there is no significant added cost. Each prize is budgeted at a certain dollar amount and nobody wants to absorb added costs.
So in your example, if the roundtrip airfare was budgeted from LA to London there would likely be no added cost (and perhaps some savings) for you to fly from New York. With a little sweet talk you would likely be accommodated. In my experience it's all on a case-by-case basis with money being the primary consideration.
Here's an extremely unusual piece of history from the 1980s: because of my relationship with one producer through working on a few of his pilots, when I won on one of his regular on-air series ("All Star Blitz") I was able to trade all manner of furniture for 2 weeks in Mexico because it represented a net savings to him. In this case the cash he was spending in his deal for the furniture was greater than the mostly-traded vacations in the more popular vacation destinations. I would have been happy to take a week in Puerto Vallarta, but as an inducement to have me give up another cash-dependent gift that was not important to me he added a second week in Mazatlan that he traded with the prize folks at "Scrabble's" Exposure Unlimited.
Alas, as there are so many fewer shows on air now there is far less trafficing in prizes. But with a smile and a good reason, such as living near New York, the departure city for your trip to London would likely be changed.
Randy
tvrandywest.com