EzekielRaiden
Follower of the Way
Do you find 5e combat lasts more than 4 rounds more often than, say, one in ten combats? More importantly, do you find this at-least-four-rounds thing happens more than ten percent of the time after the first person goes down? Because that's really the problem. After the first few levels, 5e is not so deadly that people go down in the first round very often. So you're expecting 6+ round combats, or PCs that sacrifice multiple actions/very expensive resources to scrape players off the floor, rather than push the fight to a conclusion NOW and save the dying person AFTER.I don't think it would impact things as much as you imagine...
Another option would be:
Round 1: You can move, take an action, or take a bonus action.
Round 2: You can do two of the three.
Round 3: You're good to go.
Since most times getting up would be your first priority, that is basically all you would likely do in Round 1. But, if I did this I would not allow magic/healing to speed it up.
At any rate, it would take further thought and play-testing to see exactly how much impact these types of mechanics would have.