There are two methods, one cheap and one expensive, that are used to do this in industry where UV light is used for curing adhesives and inks.
The first is called a dosimetry puck and is a squat cylinder several inches across with a photodetector and a display in it. You place it on a conveyor belt, run it under your UV light source, and read the measurement off the display. They cost between $300-600 depending on features.
The second (which requires calibration against a dosimetry puck) is called a dosimetry tag which is a self-adhesive sticker coated with a dye that changes color with cumulative UV dosage. They cost a few pennies each and come in die-cut sheets. You run one under the UV lamp and then compare its color to your calibration standard and read off the dose.