Hindu-Arabic Numerals

India obviously used the same form of numerals. Why? Because they were originally Hindu Numerals, thats in the same way slowly spread to Europe from the Arab world, slowly spread into the Arab World. The oldest definite reference outside of India came from Syrian scholar and bishop Severus Sebokht, in 662. Later on it was al-Fazārīs translation of the Sindhind into Arabic which quite possibly began to get the Islamic scientific world acquainted with Hindu Numerals. While this system began to be used all throughout the Arab World, many others were used concurrently, such as the Greek system, or other native systems.

Introduction into Europe

Just a small correction, not all of Europe used Roman Numerals, for an example, the Greek system of numeration remained widespread along the Adriatic for many centuries

While the adoption of Hindu-Arabic Numerals into Europe was slow, the earliest French manuscript they are found in dates from 1275 for example, Europe seemed an obvious place for Hindu-Arabic Numerals to spread to, however this wasn't without opposition.

Anyway, the first spread of Arabic Numerals had trickled into Spain. In the 8th Century, the Umayyad Caliphate had started there conquest of Spain, and this meant that as Hindu Numerals spread across the Arab World, it also meant they traveled along into Spain, where they were introduced by the Arabs around 900. The oldest dated manuscript which contains the numerals produced in Europe, the Codex Vigilanus (a compilation of various historical documents accounting for a period extending from antiquity to the 10th century in Hispania), was first produced in Spain in in 881, but was updated up to 976, and it was in the 976 version which contains the first use of Arab Numerals in Europe.

At the same time however, these numerals were being spread in other ways, through trade and movement between Europe, and Islamic Spain and the Levant. While this had been going on for some centuries before him, it was a later mathematician, Leonardo of Pisa (also known as Fibonacci) who had traveled the Arabic world as a merchant with his father, who directed a trading post in Bugia in modern day Algeria, it was here that he learned about that he learned about the Hindu–Arabic numeral system.

Recognising that this was easier to do mathematics with than Roman Numerals, Fibonacci traveled, studying under leading Arab Mathematicians. After returning from these travels he wrote what he had learned in Liber Abaci (1202) and did similar for Geometry and trigonometry in Practica Geometriae (1220).

While these were helping it spread into Europe, remember, adoption was very slow, and what little adoption there was, was met with opposition from the public, as they made merchant books difficult to read. This got to the point that in the statutes if the Arte del Cambio of 1299 and even later the money changers if Florence were forbidden from using Arabic Numerals and made to use Roman ones.

During the 14th century however you do begin to see some adoption of Arabic numerals by Italian merchants, including some intermediate forms, such as II III XV for 2315.

Edit: I may have to come back to this and explain the number systems in the other parts of the world you mentioned, however I will add in a bit about Chinese influence over Arabic Mathematics

Chinese and Indian Decimal Place and Position System

The combination of these two systems first appear in China, before then spreading out into other parts of the world, due to people such as merchants. There are theories (with some evidence, but very little) that from China it reached places such as the near east, and some ideas from it gradually took there place beside other, older non position systems. Later than this, it was developed independently (again theories that there was some exchange of ideas between India and China however) in Hindu Mathematics and the first reference in India appears to have been in 546, where there date 346 is written in such a decimal, place value notation. Even long before this however, the Hindus had a system of expressing large numbers by means of words through a place-value method. This included early texts with the word Sunya, meaning zero. The oldest record with a sign (instead of a word) for zero is in the 9th century, which is much later than the development of zero in Mesopotamia, however the Mesopotamian use wasn't a real zero in how we see it today, it was just a placeholder because it was not used alone, and also not used at the end of a number.

Sources

http://www-history.mcs.st-and.ac.uk/HistTopics/Arabic_numerals.html

A Concise History of Mathematics - Dirk J Sturik - Pg 67-68

A Concise History of Mathematics - Dirk J Sturik - Pg 80-81

Edit 2: Made a small mistake claiming where I linked the development in China to the one in India in the Chinese and Indian Decimal Place and Position System. Both of these systems were in fact developed independently. I meant both arrived in the near east around different times.

Arabic numerals are directly derived from Indian numerals.

This is my first answer here, so I hope the format is fine.

Arabic numerals were known as Hindu numerals in the Arab world at the time, since that's where the Arabs got them from. What we do know is that Al-Khwarizimi appears to be the first one in the Arab world to use Hindu numerals in his treatise, which was later translated into Latin in the 12th century.

In the 8th century, Islam started to gain more influence in North Africa. And Spain was conquered in the 8th century as well by the Umayyad Caliphate. The common belief is that it was through Spain and North Africa that the Arabic numeral system was adopted. Fibonacci performed part of his studies in Algeria, and is the first European that we know of that promoted and used the Arabic numeral system, namely in his book Liber Abaci in 1202.

However, they don't seem to have been in widespread use until the printing press was invented in 1482.

Couple of interesting links:

http://en.wikipedia.org/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system#Adoption_in_Europe

http://www-history.mcs.st-and.ac.uk/HistTopics/Arabic_numerals.html