BBC presenters seemed to be split evenly between Budapest and Budapesht when describing the location of a recent EU meeting in the Hungarian capital. Every Hungarian that I have spoken to (in English) uses the pesht ending. Obviously the Hungarians are "right" but either way you will be understood.

"Beijing is BEY-JHING and not Peking"

It's BEY-JING (i.e., it's not a soft French-style "j").

To make matters worse "Peking" was never pronounced the way it looks - that
was the product of a peculiar spelling system adopted by the 19th-century British Post Office to represent Chinese names. Right now I'm reading a 1950s book on the birth of the Chinese Revolution, and having to relate its spellings to names to modern-day Pinyin spelling (e.g. Nanking = Nanjing, Hankow = Hangzhou) is a pain.

Anyone who uses Rick Steves as a guide for pronunciation will be misunderstood throughout Europe.

Nuevo(a) York is New York.

I don't see anything wrong with trying one's best to pronounce place names as the natives do. To take someone to task for giving it an effort seems very petty to me.

However if I were doing a TV show like RS, I'd be sure to translate ones that are very different from the traditional English usage. Budapest/Budapesht would be OK, but if I used Firenze I'd point out that it was Florence in the English-speaking world, etc.

I think its funny that Americans say Hong Kong with the emphasis on the HONG, whereas Brits and Irish put the emphasis on the KONG.

I think you should have posted your question on the Rick Steves website.

Henry Higgins pronounced it thusly -- just make it rhyme:

"There he was, that hairy hound
From Budapest.
Never leaving us alone,
Never have I ever known
A ruder pest"

Boo-duh-pesht