Since there are no coatings on the UK's, it should be very simple to dye.

Take the time to do it right (per the instructions). Don't think that cold water or "warm" water will do when it says to use HOT water. Get HOT water.

Pre-soak the kilt in hat water before putting it in the dye. This helps the dye to be absorbed evenly to avoid light and dark spots here and there.

Stir it well to ensure every inch gets dyed.

It may even be worth adding a vinegar soak to set the color. I've never done this though.

All that being said, there are still no guarantees it will turn out like you hope.