When will the Bitcoin Cash difficulty adjust next? Bitcoin Cash adjusts its difficulty downwards if there is, roughly speaking, a 12 hour gap between blocks. At least that's my understanding.

Difficulty has dropped to about a quarter of what it was because if that and more blocks are coming in now.

Does this not mean that the new difficulty rules will no longer be triggered now? It seems difficulty has adjusted enough to never again trigger the rules with high likelihood.

That would mean that the next difficulty change happens through the regular mechanism. This regularly happens every two weeks but because the hash rate is still so low it could be far longer than that.

The last 30 blocks took about 21 hours. Normally, 21 hours would have produced 126 blocks on average. So it seems the speed we are seeing is roughly a quarter of the normal speed despite the lower difficulty.

And that seems to imply that this difficulty period will take roughly 8 weeks to complete assuming that the hash rate on that chain stays the same.

Is this assessment correct?
The original bitcoin adjusted difficulty every 2016 blocks, which would nominally be 14 days @ 6 blocks per hour.  The 238th such adjustment is due at block 479,808, and we are presently (early 4 Aug 2017) at 478,620 or 1188 blocks away on BCC.  I assume the original adjustment interval has not been changed.  35 blocks were generated in the last 24 hours, so at that rate it will take 34 days to adjust (~5 weeks).  However, miners can jump in and out of BCC according to price/difficulty ratio relative to BTC, since they use the same mining hardware and just different software.  So I expect the mining hash rate to be variable.

Note that BCC allows for larger blocks, and several have been mined that are >1 MB.  So even at 1/4 the block rate, theoretical capacity is twice that of BTC.  BCC transaction rate is running about 1/8 of BTC, so blocks are currently nowhere near full.  Since many exchanges and wallets are not yet set up to handle the forked coin, volume is likely to grow over the next month or two.