DIAGNOSIS The filter is clogged, either by sediment from your tap water or, more likely, by not being pre-soaked. The water has to force its way through dry, compacted carbon blocks.
FIX Remove the filter. 1) Give it a more vigorous prime: hold it under a strong stream of cold tap water for a full 60 seconds, rotating it to hit all sides of the mesh. 2) While the water is running into the open end, gently tap the filter against the palm of your hand to help dislodge particles. Reinstall it and try again. The flow should improve significantly.
IF THAT DIDN'T WORK If you have very hard water, mineral scale might be clogging the filter's intake pores. Try soaking the filter in a mixture of one part white vinegar to three parts cold water for 15 minutes, then rinse thoroughly under the tap before reinstalling.
HEADS UP The flow rate will always be slower when the reservoir is full because of the water pressure on the filter. For faster fills, only put a cup or two of water in the reservoir at a time.