Prior to the War of 1812 the Americans had a stronger position in Columbia/Oregon Country than the British Empire did. The lead was taken by the Astor Company out of Manhattan, trying to get a colony going so they could send supplies up to Russian Alaska, pick up a load of furs, and then take that over to China for trade. There was a promising but narrow window of opportunity for starting up a Russian+American trading relationship with Japan, but it fell through. There was some set back to the Astor Company's efforts in the region because of its remoteness, the difficulty of navigating the Columbia River, and the ecology. If John Astor had sent some better lumberjacks they probably would have had a better go of it. The Royal Navy could put more ships out there than the US Navy could so when the War broke out John Astor sold his stakes there to the HBC to get something out of it rather than having it taken away completely. With no War of 1812 the American might be able to get Vancouver Island and most of the Fraser River watershed.
As it was, I think Polk could have flexed his position a bit more, especially if he had less internal opposition and if he hadn't boxed himself in with a one term pledge. A stronger Harrison/Tyler presidency could have helped.
For the British and Canadian perspective the same is true as with Astor, if the HBC could have got more there and sooner they probably could have got Washington, northern Idaho, and western Montana. I don't think they had the means to settle all of Cascadia, so if you want them to push down into northern California towards the San Francisco Bay it will need another war after 1812. Another thing they could do is manipulate and insist that Oregon Country and the Central Valley of California become independent states, but I don't know if the British had the geographic understanding of California at that time to prompt that policy before the Gold Rush started.
I don't think the Spanish could have set up a colony there.
If Japan could have started an earlier Pacific Empire, they could have used Hawaii as a spring board to get to the PNW, but I don't know what there would be to draw them there that they couldn't also have gotten closer to home.
If a potato farming culture had started up in the PNW before the 1700s, that would be intriguing. And if Europeans had saw an intense potato cultivation in another locality the practice might have spread to other places around the world quicker, such as going to the Dutch and Huguenots in South Africa.