How can I photograph a nighttime glow run without a UV flash?

With the constant UV lights providing the illumination it's just a question of getting the exposure right and limiting camera movement. You're probably going to have to accept some motion blur from your subjects but that can potentially add to the mood of the photos.

Unless there are some significant light sources outside (i.e. overhead street lights) you're going to have to use flash or some other form of artificial light to get anything. If possible set your speedlight to E-TTL and dial in a stop or two of negative flash exposure compensation. Use second curtain sync if possible. Turn on AF assist if it is available with your camera mounted trigger or off camera flash unit. An alternative to the wireless triggers if they don't offer E-TTL, second curtain sync, or IR assist functionality is to use an off shoe flash cord. Before getting a set of YN622-C triggers I sometimes would use an off-shoe cord and would hold the flash out to the left with my left hand while controlling the camera with my right hand. I even developed the ability to turn the zoom ring on a few of my lenses with the middle finger of my right hand. It takes a little practice learning how to angle the flash without looking at it for every shot, but it works fairly well and allows you to get the flash further off the optical axis than most brackets will allow.

Inside with the constant UV lighting, if you choose not to use your flash, you'll probably need to set exposure manually and will definitely need to focus manually as well. And be sure to save your photos as raw .cr2 files! It will give you much more latitude to make them look the way you want in post.

The following was taken with a Canon EOS 5D Mark II and an EF 24-105mm f/4 L IS lens set to ISO 6400, f/4, and 1/4 second (IS on). If I had know the black lights were going to be on when I walked into the gym for the pep rally I would have had the EF 50mm f/1.4 mounted on the camera instead and had my monopod with me. I took a handful of shots and then changed the lens but by that time the normal lights had been brought back on. I could have used f/2 and 1/15 second which would have reduced most of the blur from the camera movement (Sometimes I'm pretty good hand holding. Here I was leaning back against a wall for extra stability), but very little of the subject motion blur.

In post I bumped up brightness 1/3 stop, reduced contrast and then pulled down the blacks to help with noise and posterization, and bumped the color saturation up a little to add to the surrealistic effect. I used a color temperature of 7000ºK, but there was no real difference anywhere above about 5400ºK. The lower you pull the color temperature, the more saturated the purples and blues will be. Depending on your exposure, the purples and magentas will start to blow out somewhere around 3000ºK.