Written, yes. The shapes of hangeul have remained consistent over the years. The written words would still be intelligible across generations.

"Hangeul" refers only to the written language, however); in that sense it can be considered like the English alphabet, in that one doesn't speak Alphabet. "한국어" -- loosely, "Hangook-uh" is the spoken Korean language. I'm unqualified to comment on the differences between modern Korean and classic Korean, though.

It merits mention that hangeul is shockingly easy to learn. When I lived there I picked up hangeul in a morning and was comfortable sounding out text by the afternoon (though comprehension is obviously a different story). Sejeong's mission was to create a simple written language that would serve as an archetype for the spoken language and increase literacy by means of a simplified writing system (as contrasted with hanja), and to that end the mission was quite successful.

Could a modern Korean speaker, if time traveled into historical Korea (pretend you're a consultant for a K-drama!), be able to effectively and easily communicate with the people of the time?

If the modern speaker spoke Chinese, maybe.

Korean used to be tonal and Hangeul denoted this with dots on the left side of the character. The tonal properties began to disappear in the 16th century and then disappeared completely in the 17th century. Today, Korean is really tone deaf but there are still some vestiges that remain. Homophones can be distinguished through longer/shorter pronunciations although this is generally not enforced/followed. When it comes to reading, Korean newspapers (and other publications that need specificity) will use Chinese/Hanja characters if the meaning would be unclear.

Further, there are some Hangeul characters which are no longer used. Some of these characters represent sounds that are not used in modern Korean, the main one representing the the "z" as in "zoo" sound. The other characters that have disappeared were very similar to other characters that still exist.

So much of Korea, including the language, was influenced by China. The tonal system and the deprecated characters would have been critical at the time, but have fallen out of use as Korea became culturally more independent. So, could a modern Korean speaker be able to communicate with a 15th century Korean? Probably, with some help, but another critical thing to look at is the word choice. We hear about it today - North Koreans have trouble understanding South Korean because the language has changed so much - but it's a big issue over time, too. Korean is to Chinese like English is to Latin/Greek/German. For every word, there is a Sino-Korean word and a pure-Korean word, but many of these pure-Korean words are of newer origin. Most educated Koreans of today would know both words, but pure Korean sentences are very different from Sino-Korean sentences. A modern speaker would have to consciously choose Sino-Korean words instead of pure Korean words to communicate with someone from 500 years ago.