Anthropology here, I recall reading something about this in the book Tales of a Shaman's Apprentice.

The writer was in the South American jungle talking with a clergy man (iirc) whose hobby was to listen to stories from members of the various tribes he came in contact with. These tribes had rarely (if ever) had any contact with the outside world and so their stories were through oral tradition, passed on verbally from generation to generation. According to that man one of the more memorable ones he heard was a tribal legend about a time when their ancestors had traveled long distances hunting large animals and had to cover their entire bodies in animal skins to keep from dying.

The author speculates that if that story held true it could have been an actual oral tradition of the crossing of the Bering Strait, preserved through storytelling. So, in hot, humid environments like the Amazon the concepts of snow and ice may have been lost through time but actions their ancestors had taken to survive it could have stayed in the stories. They would have at least had the idea that some factor had been there that necessitated those steps even if they didn't really understand the meaning or purpose behind it anymore.

Edit: I want to point out it was the author's speculation, not mine. As much as it's an amazing idea to think the crossing could have been retold for that long, it's also unsubstantiated.

As /u/betacyanin points out, oral tradition and story-sharing certainly would have spread knowledge about snow, ice and cold far beyond the geographic areas where they might be found, even in prehistory. We know from archaeological work that trade networks have existed for much longer than writing has, which provides a conduit for such stories.

Furthermore, as you probably know, snow is not limited to high latitudes. It can be found at high elevations as well, including near the equator. In the Americas, the high mountains of the Andes are snow-covered year round. Similarly, the high mountains of east Africa, in particular Kilimanjaro, are very nearly on the equator. I can't speak to Singapore and southeast Asia, but I will talk about the African experience. The writings of Johann Ludwig Krapf, Hans Meyer and Johannes Rebmann are of particular interest.

Writers as far back as Ptolemy reference a "Moon Mountain" in Africa, and it seems logical to think that Kilimanjaro's snowcap would be compared to the white of the moon. Travelers to Ethiopia and the East African coast also record references to the mountain or mountains of the Moon. Toward the middle of the 19th century, a pair of German missionaries set out to find these rumored mountains and became the first Europeans to definitively sight Kilimanjaro.

On May 11, 1848, Rebmann wrote in his diary:

"This morning, at 10 o'clock, we obtained a clearer view of the mountains of Jagga, the summit of one of which was covered with what looked like a beautiful white cloud. The guide called it merely 'cold,' and at once I knew it could be neither more nor less than snow."

Rebmann and Krapf weren't initially believed by everyone, particularly in England. In that country, a famous geographer named W.D. Cooley said it was impossible for snow to exist in equatorial Africa. Rebmann's sighting was "a most delightful mental recognition not supported by the evidence of the senses."

Krapf attempted to rebut Cooley's arguments in his book, and you'll have to excuse the extended quote here:

"And why should the existence of snow in Equatorial Africa be doubted when there is snow under the Equator in America; when there is snow sometimes on the Cameroons of Western Africa; when snow is frequently observed upon the heights of Mount Amba Hai in Abessinia, being situated 16,000 feet above the level of the sea; and when there is snow in the south of Kaffa as Bruce has positively stated?

On my first arrival at Mombaz in 1844, I had not the least idea of the "silver matter," as it was called, reported to exist in the interior, being snow. I regarded the silver and the dshins or evil spirits of Kilimanjaro as some kind of impalpable white sand, which, as the Arabs told me exists in Southern Arabia, where the sandy baher-el-safi forms a kind of whirl-pit swallowing up everything that approaches it. I was not then aware that Ptolemy had alluded to a mountainous country, and that these mountains were covered with snow, and considered by him as the real sources of the Nile."

Cooley's position was disproven by further observations announced the same year Krapf published his book, and several attempts were made to climb the mountain between 1861 and 1912. Hans Meyer's 1891 book, An Account of the First Ascent of Kilima-Njaro is particularly interesting reading, though like all primary sources it must be taken with caution.

Singapore is located on a major trading route and has had visitors from present-day India and China for millenia, who would surely have been familiar with snow. The earliest Hindu texts do certainly mention snow, see for example Hiraṇyagarbha Sūkta from the Rig Veda.

Also of note is that snow and ice can be found on mountaintops not terribly far from Singapore - mount Kinabalu in Borneo, for example, and in the mountains of present day Thailand and Myanmar.

So in the Malay peninsula, at least, people who met travelers or locally residing Hindus and Chinese would have heard about snow.