I can't offer the global answer. However, I can offer you a piece of the puzzle in the example of the adoption of Western mathematics in Japan.

Basically, Meiji Japan adopted Western style mathematics because it was the style of mathematics already in global use in areas of science and technology that Japanese leaders felt were necessary to strengthen the country and protect it from the kind of imperialist incursions China had suffered. It is telling that the few members of the Dutch Studies school in Nagasaki who studied Western mathematics seriously before the Meiji Restoration were primarily interested in the ballistics applications of calculus.

To contextualize, native Japanese mathematics had followed the Chinese style of calculation, and primarily used an abacus (soroban), rather than notation on paper. When they did write mathematics down, they used the kanji numerals. Although Japanese mathematics had the basic tools of algebra (including notations for unknown variables and ways of representing zeros), it lacked the concept of a function that was so critical to calculus. Moreover, there were no abacus algorithms for ratios or proportions, which people had been forced to do mentally.

Given the importance of those concepts to much of the Western technology that was enabling imperial expansion, Japanese leaders pushed for the adoption of Western mathematics to facilitate Japanese use and development of similar technology. Adopting western notation was part of that effort. In 1855, shortly after opening ports to more Westerners than just the Dutch, the Tokugawa Shogunate set up a naval academy in Nagasaki that taught Western mathematics, using the existing knowledge base from the Dutch-studies scholars. After the Meiji Restoration, leaders pushed for an aggressive reform program under the slogan "rich country, strong military." Those goals required a technologically literate populace. In the interests of creating one, the Meiji government made basic education compulsory in 1872. That curriculum included Western mathematics, taught with arabic notation.

In short, arabic notation was what the industrialized people with the stronger guns, steam ships, and trains were using. Joining an ongoing global scientific discourse, Meiji leaders felt they had best have their people learn that notation system if they wanted to use those people's books to make their own factories, get some faster transportation, and have more powerful guns.

However, this doesn't explain at all why the earlier industrializers, like the British, were using arabic notation in the first place. Hopefully someone with more knowledge of the premodern history of science will be able to explain that.

Edits: Just fixing my numerous typos and circuitous sentence structures.