The chilli pepper is very popular in Europe and has been for quite a long time. The German Pusztaschnitzel, Spanish pimientos rellenos and Italian 'nduja all contain peppers from the Capsicum family, not to mention the adoption of Indian and Far East Asian foods in British cuisine. Paprika-flavoured snacks are very popular in Eastern Europe - paprika being made from dried peppers. The Icelandic people are fond of liberally seasoning chips (err... "French Fries") with paprika and garlic salt. It's delicious!
The issue here is when chilli peppers became a feature of European foods. According to this colourful book, natives of the Caribbean were rather fond of chilli; samples were brought back home for the Spanish nobility to sample in the late 1400s by Diego Chanca. Spicy sausages were a feature of Iberian cuisine and chilli slotted nicely into their repertoire; though those pesky Catholic priests stressed the dangers to the layperson of becoming over-passionate and excitable due to chilli consumption. Spain wasn't immediately enthralled but the chilli had arrived.
Europe isn't ideal for the cultivation of chillis. They behave in a similar way to tomatoes, another New World fruit, enjoying strong sunshine and are quite thirsty. Growing profitable quantities necessitates the use of large-scale greenhouses, but these are a relatively recent invention with early 17th Century Dutch experiments proving hard to seal against the elements. It was more efficient to grow in quantity where the environment was more amenable. The hot and humid conditions of Indochina were an ideal space.
Helped by the growth of Portuguese trade, the chilli pepper was introduced to Goa and spread across South-East Asia. There is no adequate explanation of their sudden and indomitable popularity in Indian and Thai cuisine. We can only speculate: perhaps it was the variety in colour or the almost painful levels of spice that capsicum can produce that made chillis so sought after. Their growth to being virtually omnipresent took place over a remarkably short period of time and by the 19th Century, the chilli pepper had made its position permanent.
Edit: Corrected outdated German words.
Second edit: With thanks to /u/EvanRWT:
Chili peppers grow in every single state in India, and the tropical climate often allows multiple growing seasons every year. They are also extremely cheap and easy to grow, unlike black pepper, which takes a good deal of work. There is no processing involved. Most Indians eat them raw as a condiment, or just add chopped fresh chilis directly to their food. And it's trivially easy to store them - just spread them out to dry in the sun for a couple days, which is all it takes in the tropical sun. The dried peppers will easily last a year.
That doesn't fully answer your question but the source listed in the second comment should answer your questions pretty well.
How long has the chili pepper been used in China & India? Did they develop the plants independently, or were they imported from South America? What about Africa?