I live outside Seattle WA, your chances of see snow in vancouver BC that time of year are low, but it’s going to be mostly cold, windy, and rainy the entire stretch from there to Seattle. South of Seattle, unless you take the coast roads, it’ll be cold, but drier and less windy till northern California. Between southern Oregon and northerner California you might run into some snow depending on route, and how long it takes you to get there, but not likely if you clear the area by Jan, and very unlikely to collect on the road. Rain and wind are going to be the big issues for you, and generally chilly in the 5c to 10c range. It’ll stay a little cold all the way till central California (San Francisco area), but less rain the further south you get. After that winter tends to be pretty mild, mid-February in that area might be 15c, but often sunny.
If you plan to head up into the mountain ranges this all changes, by October the snow collection is pretty substantial and small highway passes are closed by then. Even though it’s June now there are still lots of high elevation roads have snow on them.
But, if you’re just doing north -> south and staying near the coast you’ll be fine, just dealing with the cold and wet.
All that said I’m sure it’s hard to adjust your schedule, but personally I’d move it a little if you can. Either start in South America in October when it’s spring, or start in North America in April which is spring. Of course ending up in Central America in summer is also not great. This route over six months basically means except for Central America it’s going to be winter pretty much the entire time. So either you can be cold for half of it, but rarely hot, or hot for half of it and rarely cold.
I’ve been all through Chile in June and the southern half is basically western Washington, western British Columbia in November. I was annoyed I went in the middle of my summer

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My wife, back to the driving rain, appreciating that I took her to Chile in the dead of winter.
When you come through this way hit me up!