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In this case the threat of the Zoological Gardens authorities that if
the men "came out" the animals should come out also had intensified
and precipitated the crisis. The imminent prospect of the larger
carnivores, to say nothing of rhinoceroses and bull bison, roaming at
large and unfed in the heart of London, was not one which permitted of
prolonged conferences. The Government of the day, which from its
tendency to be a few hours behind the course of events had been
nicknamed the Government of the afternoon, was obliged to intervene
with promptitude and decision.

from The Unkindest Blow, a short story by Saki

Are "conferences" the Zoological Gardens authorities' or the Government's?

Prolonged conferences here just means people having serious discussions for long periods of time, not the well-organized and extensively scheduled meetings that people mean by conferences today.

From the OED:

Conference: The action of conferring or taking counsel, now always on an important or serious subject or affair.

Presumably these conferences would be people in the government meeting and trying to decide what to do.

This meaning is more or less obsolete today.

Because he says "prolonged conferences" and not "the prolonged conferences", the author does not mean any specific conferences, but just conferences in general. Since he doesn't mention any conferences in this passage, we can assume he is talking about any and all hypothetical "prolonged conferences", which could possibly be organized either by the zoo, or by the government. i.e. There is no time for long discussion during this dangerous situation.

The author's use of "conferences" is alluding to communication (or lack thereof) and saying the members of whatever group he's addressing didn't talk to each other. Those in question essentially ignored each other, went about their selfish own tasks and therefore found no reason to be fully engaged to handle the business at hand. In other words, they may have met for some reason but since they refused to actually converse just walked away after a short time ... an afternoon. Apparently the author felt such short time in an afternoon is far too short to accomplish anything meaningful. "Government of the afternoon" is a derogatory comment critical of the group not being responsible in doing their jobs. They cared so little they may as well have met only in the afternoon and then only for a short non productive period of time.