Interesting contraption. One thing you might have to watch out for if using a conventional lathe leadscrew and worm gears is that the gear teeth might tend to ride up over the thread on the leadscrew as the skewers are only sat in slots, not retained in bearings. The pictured right-angle skew gearing puts force on the skewers largely axially, not radially like a worm and wheel arrangement would. But enough weight in the skewer handle might do the job.
There are a lot of gears and worms and wheels etc for sale cheap on Aliexpress. You might find something like that arrangement if you search around enough. They are very hit and miss with terminology so its a bit of a treasure hunt though. But to buy that many individual gears will add up.
To machine that long helical skew gear in the video would be quite an undertaking. Looks like it is 6 or 8 teeth or so. Would need to be done in a milling machine,