First things first - DON'T cut all the wires where you show in your picture. You will lose valuable cable length.
Normally manufacturers use molded on plastic strain reliefs, but I guess the chairs are low volume hand manufacture and reasonably customized. Luckily it isn't the case here anyway. That strain relief is known (in the UK at least) as a sleeved grommet, but also known as strain relief grommet, or extended grommet. They are normally specified by panel hole diameter, but will should include a maximum cable diameter too. If you buy one based on the hole diameter in the box, the cable diameter is almost certain to be correct. Yes you can get them from Ali, but you can also get them from your local electronics distributors (presumably Digikey, Mouser etc.). Depending on how quickly you want to get your chair up and running again, an ebay seller with local stock might be your quickest+cheapest option.
Now the cable...
- Yes, if you cut the cable tie, the cable should slide out. The reason I said don't cut the wires where you indicated is that there is probably a significant length of wire internal to the joystick that would otherwise have to come from your remaining cable, once cut.
- You haven't shown a photo of how the wires are terminated inside the joystick. They will either be soldered directly into a PCB, go to a connector, or maybe a screw terminal strip arrangement. If they are soldered into the PCB, the easiest way is to unsolder them and re-solder when you have replaced the grommet. If you are not happy with soldering to the PCB, the simplest solution is to cut the wires a reasonable distance from the PCB and splice them after. Be sure to document (photo) where the wires go anyway. Leave the blue wire attached as it's going to need splicing anyway, just pull it through the remains of the cable and grommet after you remove the cable tie.
- Once you have all of the wires disconnected inside the joystick, you can cut the cable tie and pull out the cable. You will be left with a stub of cable outer on the end, but the wires should pull out of this (slit if necessary).
- You can then remove the grommet and fit the new one. Slide the wires in and push the full cable diameter through the grommet. Push it far enough that you have undamaged portion to (tightly) fit a new cable tie. You should now have plenty of slack available on the wires inside the joystick. Depending on what you did to disconnect them, either resolder or splice to the ends inside the joystick. If you are splicing, make sure you get some heatshrink sleeving to insulate them. The other wires may have been weakened by flexing at the point where the blue one failed, but as long as they're still reasonably ok, they won't be under any further stress once they're inside the joystick housing.
- Finally, add a couple of blobs of glue at the cable grommet inside the housing. One at the cable tie to stop the cable rotating inside the grommet, and another to stop the grommet rotating inside the joystick housing. You might want to put the joystick in your normal use position and adjust the rotation first, before gluing, to make sure you don't have any annoying twist in the cable.
You could probably do with a cable tie or something somewhere on the frame to prevent the joystick from falling to far and getting mangled again. If you post a photo of the wire terminations inside the joystick we can probably advise further, but hopefully I've covered everything.