HDPE does not play well with ordinary glues (PVA, CA, epoxy, and polyurethane). In fact, I use HDPE in my shop to make jigs and fixtures where I specifically don't want glue to adhere to it. There may be specialty products that are made specifically for HDPE, but I'm not familiar with them.
That said, I have had some limited success in gluing HDPE to wood by using a creating an 'interlocking joint'. For example, I have a live center that accepts interchangeable 'noses', and I've made a number of wooden noses with HDPE faces or points that I can screw onto the live center. In each case, it was necessary to glue a bit of HDPE into a recess in a turned, threaded wooden block. The way I've done that is to turn a scrap of HDPE (salvaged from a cutting board) to fit snugly into a recess that I drilled or cut in a turned wooden block. Then, I've cut one or more grooves into the HDPE. Then, I apply glue liberally to the inside of the recess and to the outside surface of the turned HDPE, making sure that the groove(s) are filled with glue, and then press the HDPE into the recess into the wood. When the glue cures thoroughly, the glue in those gooves forms hard plastic ridges that are tightly bonded to the wood. While the glue doesn't actually adhere to the HDPE, the ridges mechanically lock into the grooves in the HDPE, creating a mechanical attachment rather than a true bond. I've used PVA and epoxy in this way, and I suspect that polyurethane would also work (provided the glue up is clamped so that the expansion that takes place as the glue cures doesn't disturb the physical positioning of the components). CA might be a problem because it is notoriously brittle after it cures, and the mechanical joint could more easily fracture than would a mechanical joint using glues that are more flexible when dried.
I suspect that you could do something like this to glue tubes into HDPE blanks, but it would be tedious since the groove in the HDPE would have to be inside the hole.