All 3 Uses

consequence
in
A Tree Grows in Brooklyn
(Edited)

Consequently, the four daughters grew up having little communion with their father.
Chpt 7

consequently = resultantly (as a result)
Consequently, there was a great surprise waiting for Johnny when he sauntered home to get a fresh dicky and collar for his evening's job.
Chpt 14 *
What she wrote was of little consequence.†
Chpt 26 *

consequence = importance

Definitions:

(1) (consequence as in: a direct consequence of)
a result of something (often an undesired side effect)
(2) (consequence as in: of little consequence)
importance or relevance
(3) (meaning too rare to warrant focus)
In classic literature, consequential may refer to someone with too much feeling of self-importance as when Dickens wrote "Because he's a proud, haughty, consequential, turned-up-nosed peacock."

Self-consequence was used in a similar manner, but is more easily understood by modern readers since important is one of the modern senses of consequence.

Another classic sense of consequent that is similar to importance or significance refers to "material wealth or prominence" as when Jane Austen wrote: "They had each had money, but their marriages had made a material difference in their degree of consequence."