Instantly is probably a bit optimistic. The bacteria are going to distribute themselves all over the different tank surfaces -- including the insides of both filters. They prefer the filters because of the fluid flow -- the flow brings more food than non-flowing water. But, both filters will have a fair number of bacteria in it. When you take one out, there will be less bacteria in both tanks. Both tanks may undergo a small mini-cycle for a day or three or so.
In cold water, the bacteria won't reproduce as fast, so to make up for the loss of one filter, there will be a lag as the only remaining filter catches up. It would probably be worth taking a few precautions/checks right before making the switch. Scheduling the day of the switch as a "fast day" where you don't feed the fish that day will help lower the amount of waste. And, just be ready to go a water change should the level of ammonia be high when you check it. As I wrote above, it will probably be fine in a day or so.
In the new tropical tank, the issue is that the bacteria were cultured in the cold water. They will have to transition from being cultured in cool water to warm water. This will take some time, again, a day or so. The good news is that once they are used to the warm water, they will reproduce quickly. Again, however, their numbers will be a little low. Like above, be ready to test for a few days and do a water change if needed to dilute the pollution.
All in all, everything should settle back down in just a few days. Definitely better than than a month's length of time it would regularly take to cycle. But, it isn't "instantaneous". Maybe I'm just picking nits, but I think that it is appropriate to be as accurate as possible because it is caring for another living creature's well being.