Noone Special - they way I was tought in 7th-grade Arabic, Al-Quds is short for Urushalim Al-Quds, or “Jerusalem the Holy.” Think of it as a nick-name.
Actually, it’s pronounced Veen, not Ween.
I’m told Puerto Rico is referred to as Borinquen among many of the islanders; apparently it’s an adaptation of the original native name of the place. I don’t think that name has any official status though, so it might not count.
Auschwitz is the German name for the now Polish town of Oświęcim. Since it probably was of little interest to English speakers back when it had a significant German population I suppose it does fit with the OP, but the name was once used by the inhabitants.
In the Republic of Ireland, IIRC, road signs have English and Irish names for cities and towns on them. It seems like virtually all large towns and cities in the Republic of Ireland have been renames by the British and the names are still in place. Wikipedia has lists of Irish towns and cities with their original names.
Across the Irish border, in Northern Ireland, a lot of fuss is made by some to have both the English and Irish names of streets on signs.
Bit of a stretch, surely? There’s only 60,000 Gaelic speakers in Scotland, out of a population of over 5 million.
Teegeeack/Earth?
whaaat???
The OP asked for the ‘local language’ - without defining it more clearly (note my comments about other Celtic languages 
)
This is seriously in danger of getting into the neverending debate over Jerusalem’s status, but even accepting Israel’s control over it, Arabic is common as a native tongue among a large percentage of the city’s population, and even more, it’s an official language of the State of Israel. Which makes me wonder - maybe you can help here - what Jerusalem is called in official Israeli documents in Arabic.
This also holds true for many of the place names within Wales; some of them are recognisable English versions of the name (Caerdydd=Cardiff), others are literal translations (although as very many Welsh place names are literally descriptive of features of the the location, the correlation may be coincidental) and yet others are just completely different (Abertawe=Swansea).
Groovy- in that case, let me add one more:
United States- Turtle Island
I haven’t seen the Thai name for Bangkok yet given – Krung Thep (กรุงเทพ).
Erk. In Thai letters, that’d be กรุงเทพ .
I’m aware of this. But Auschwitz is the name it is known by in America, at least, so I felt it fit.
An important locus of Indian civilisation were settlements on a river (now almost entirely within the borders of Pakistan), known locally as Sindhu, from a Sanskrit word meaning something like “big water.” Persians translated Sindhu as Hindu, and the Greeks took the Persian form and made it Indus. Alexander’s campaigns ended somewhere in the vicinity of the Sindhu/Indus, so the Greeks were familiar with the river and the land beyond it, which they called India. The territory around and beyond the Indus was known to the Greeks, Romans, and later Europeans as India from then on.
It’s a little more tricky to come up with the origin on the other side. The land we now call India was not politically united until the British came along. There were times when one occasional emperor or another managed to conquer most, but not all, of the Indian Subcontinent, but so far as I know, no such regime ever imposed a single name on the resulting political unit. The people of India knew about various smaller units, which tended to change over time, but didn’t come up with a unitary name.
The Turco-Mongols, under leaders such as Babur, learned from the Persians that the people over there around and beyond the Hindu river were Hindus, so they called it Hindustan, but this was never a universally accepted (by the natives) name for the entire subcontinent. The use of Hindustan as a synonym for India by Indians is a modern development and has more to do with the existence of a neighbouring state called Pakistan.
It seems to me that what happened was that, once unified under British rule, the Indian people began to form a unitary identity and then had to reach back into ancient mythology to come up with a native name. Bharat was a mythological ruler of roughly the territory comprising British India. The land ruled by Bharat is Bharat Barsha or Bharatvarsh.
This may be a bit nit-picky, but not all the English names in use are the result of renaming by the British (English). Many of the English names were conferred by the Norse founders of the towns - Wicklow, Dublin, Waterford, Wexford.
In fact, there are many placenames (about 14% of all townlands) where the Irish name is a translation from the original English. English has been spoken in Ireland for a very long time.
The biggest town I can think of at the moment with an English name is Newbridge.
Pit-nick away, I never knew that. I assumed it was all done by British rule.