Cut the 1-1/4" PVC pipe into sections of length 24", 24", 6", 12" and 3". Cut a 1/2" hole 2" down in the 12" piece, then cap the end. This piece will become the mouthpiece.

Attach elbows to both ends of the 3" piece of 1-1/4" PVC pipe. Attach the 6" piece to one end. Attach another elbow to the 6" piece, and attach the 24" piece to that. This is the body of the instrument.

Cut the 1" PVC pipe into sections of length 24", 3" and 4". Cap off one end of the 24" piece, and attach an elbow to the other end. Attach the 3" piece to the elbow, and the other elbow at the end of that. Cheaply tape the other 24" piece of 1-1/4" PVC pipe over the 4" piece of 1" PVC pipe. This will be the slide.

You should now have all the basic pieces.

If you put them all together now, it should be playable.
Complete (untaped)

However, you should apply tape liberally. The following picture shows where tape should be. Note particularly the single piece of tape around the slide. It's important that the slide be as airtight as possible, so apply just enough tape that it will impede airflow, but won't prevent you from sliding it.
Taped pieces

Here's the complete trombute, being played:

And here's a video of me playing it poorly: