Both Uses

consequence
in
Childhood's End
(Edited)

Though it was much the largest, the Freedom League was only one of the organizations that opposed Karellen-and, consequently, the humans who co-operated with the Overlords.
Chpt 3 *

consequently = resultantly (as a result)
The fact that it was a mind not yet born was of no consequence, for Time is very much stranger than you think.†
Chpt 18 *

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."