Welcome to the world of thermal dynamics. Heat RISES so it will end up on the second floor first. Even if the downstairs were blowing heat, it would still end up upstairs unless there are doors at all the stairways. The only way to convince it to go downstairs is to have the "cold air returns" at the floor level of the first floor so the heated air has to come down to be recirculated. You normally have the same problem in the summer time, but then it is because the colder air likes to stay downstairs, and let the warm air go up to the second level.
Heat doesn't rise- it moves from hot to cold.
Sure hotter air is less dense than colder air and unobstructed undisturbed air will stratify with the warmer stuff on top, but this buoyancy/convective effect is several orders of magnitude lower than the blower & duct impedance issues. (It takes a heluva hot-air balloon to overpower a 300-600W blower-motor!)
In the summer upper floors tend to have more direct solar heating plain & simple, they're, less shaded, with a 130F attic instead of a 80F room above it, and an 80F rooms instead of 70F earth-coupled basement below. It has very little to do with the density of the cooler air falling the first floor and the hotter air rising. The direct cooling load on the upper floors is simply higher. With the same duct-impedance issues as during the heating season the first floor may stay cool enough since the load is dramatically smaller. It could be that the duct system was designed to optimize the cooling balance (higher flow to the second floor), which will put it out of balance toward the second floor in heating mode.
But in the meantime if there's at least some flow out of the first floor ducts you may be able to temporarily balance things somewhat by closing doors restricting flow to the overheated rooms, and leaving the doors to the colder rooms open to minimize the return impedances.