Does the new moon appear at the same time around the world or in one region last My query is basically, say for example if the new moon is seen in the Middle east region, does it mean that it was the day of new moon in sub continental country as well ? You can't have a new moon in India for example the next day scientifically, right ? Please help me understand this with diagrams so I can explain it to others. 

To me it doesn't make sense that we will have two consecutive new moons in the same region of the planet. New moon in middle east but new moon in India is on the next day.

I am aware since it's a broad topic so timezone can change the answer so in this question. I am restricting my question to the sighting of the new moon in the same region as in say Moon is sighted in Dubai, doesn't it make sense that India will also have the first day of moon on the same day given that it all falls in the same plane.
If the new moon is measured by moonrise then it's possible that it could land on a different day in different longitudes.   Or, if it's not measured by day, but by astronomical position, it's still possible that it would fall on a different day in different timezones.    180 degrees of longitude should equal roughly a 50% chance of landing on one day before or after (2 days apart isn't possible).

30 degrees of longitude should be roughly 1 chance in 12 of landing on a different day.    More detail is in the possible duplicate question. 

Map of timezones.  While the lines aren't precisely straight, each line should correspond to about 15 degrees longitude.