I wrote to him as lately as last week.

The other options are incorrect because they do not mean “recently.” The word “late” means “after the expected or usual time,” so it does not make sense to say that the speaker wrote to the person “lately” if they wrote to them last week. The word “early” means “before the expected or usual time,” so it also does not make sense to say that the speaker wrote to the person “lately” if they wrote to them last week. The word “immediately” means “without delay,” so it does not make sense to say that the speaker wrote to the person “lately” if they wrote to them last week.