The stock Druid in DW gets shapeshifting powers straight out of the box, and also can gain access to Cleric casting by using Hunter's Brother to take the Ranger move God Amidst the Wastes. However, I have been able to find nothing in DW that specifies that a shapeshifted PC loses the ability to cast spells for the duration. Am I correct in my inference that DW druids who have the ability to cast spells can cast their spells no matter what form they're in? Or is this pure DM-ruling territory?
It's squarely in table territory.
When shapeshifted, "you still use your normal stats but some moves may be harder to trigger--a housecat will find it hard to do battle with an ogre." (DW p.107, "Shapeshifter")
So the question in your scenario is "can my shapeshifted form trigger the move cast a spell?"
Remember that your moves are triggered by narrative: if what your animal form is doing in-fiction is unleash[ing] a spell granted to you by your deity, then you cast the spell.
This then brings us to p.18: "Making Moves."
If it’s ever unclear if a move has been triggered, everyone should work together to clarify what’s happening. Ask questions of everyone involved until everyone sees the situation the same way and then roll the dice, or don’t, as the situation requires.
While the Druid is shapeshifted,
You still use your normal stats but some moves may be harder to trigger—a housecat will find it hard to do battle with an ogre. The GM will also tell you one or more moves associated with your new form. Spend 1 hold to make that move.
So, unless the Druid has chosen a form that gives them a Spellcasting move, they need to trigger the Cast a Spell move normally. This is where I, as a GM, would default to Ask Questions and Use Answers. Ask the Druid (or the Cleric, if there's one in the party) what unleashing a spell granted to you by your deity looks like. Does he brandish forth a holy symbol? Intone a passage from scripture? Hold forth a hand and let holy light spill from it?
Once you know what triggering the move looks like, it will be very easy to judge whether or not a given animal form is capable of Casting a Spell.