According to dictionary, one definition for a intransitive use of the phrase "pick up" is to improve. So, could I write these?
- The situation finally picked up.
- His health finally picked up.
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The first one is a bit tricky. It's hard to tell what about the situation is picking up; pick up can also mean improve in speed or grow stronger:
The winds have really picked up out there.
In this case, you're probably better off just sticking to "improved."
As for your second sentence, it's perfectly acceptable, though I do think you're more likely to see "improved" when talking about health.