DIAGNOSIS: The new filter cartridge isn't fully primed with water. Air trapped inside creates a lot of resistance, and the machine's safety logic slows the flow to a crawl.
FIX: This is normal for first-time setup. Take the filter out of the unit. Hold it under a running tap for a full 60 seconds, rotating it to wet all sides. You'll see bubbles coming out. Shake out the excess water, reinstall it firmly, and fill the external reservoir. Let the machine sit for 15 minutes before trying to dispense. The flow should be normal.
IF THAT DIDN'T WORK: Check that you removed all the packaging. There's sometimes a small plastic shipping plug or tape over the water inlet inside the filter compartment that's easy to miss.
HEADS UP: Never use a generic or off-brand filter. The flow rate and pressure are calibrated for the specific Brita cartridge, and using others can damage the pump or void the warranty.