Current timeline is for getting all 4 cameras to work, but I think only one needs cryogenic cooling, other 3 could work with just sunshield. In another answer one of the engineers said that first images will be released once telescope is fully operational (aka all 4 instruments cooled down).
Probably hard to resist the temptation:)
There might be good engineering reasons to experiment early, like testing comms systems etc
Here is first technical image from Hubble, before its mirror's flaw was discovered: https://stsci-opo.org/STScI-01EVTBEME0059ASGX8X3DFWZMX.jpg
> Early Release Observations (EROs) will be taken by JWST during both the commissioning and post-commissioning phases of operation. These observations will be chosen to have wide public appeal and are designed to demonstrate the capabilities of the JWST instruments. Publication or reporting in any form of results of these observations is embargoed until the EROs are released.
https://jwst-docs.stsci.edu/jwst-opportunities-and-policies/...
There is a chance for first images to be released before commissioning is complete.
Then again, it's not a unique situation for an engineer. It's the same for any situation for construction (and related jobs) estimates, auto-repair, etc. You always want to lean on over estimating time and finishing "early" vs the opposite.