I've only been to Jeju once in the winter (a couple of years ago in late January/early February) and I spent a couple of days on different sections of the Olle Trail on the south coast. Both days it was sunny, although quite cold, but there was no snow anywhere. I think when Jeju gets snow it tends to be up on Hallasan, rather than on the coast.
In total, I've hiked parts of probably 6 or 7 sections of the Olle Trail and I really like it. Each section I've been on seems to have a lot of variety, and although I'm sure some parts get busy, none of them have ever been busy when I've been on them. The bus system is also quite good and there's usually an announcement (also in English and sometimes also in Japanese and Chinese) for bus stops that are at the beginning/end of a section.