There are 2 C clips holding the cable outer to the chassis in the black hole. You can get those off with a long screw driver.
Once you have released those two, and the inner cable ends at the drums, the cable can come forward to allow you to get at the horseshoe. Then with the horseshoe up on top of the hole, use a drift to force the end jaws apart so the cable can come out of the horseshoe.
Use copperslip or grease on the inner where it engages with the horseshoe.
When you put it back together, make sure the thread on the adjuster nuts is cleaned up so the nuts run freely. To get the spring back into place, put a loop of wire through from the engine side to pull the C on the end of the spring into place.
Good luck with getting the C clips back on the new cable outers in the hole.
Good luck with adjusting the cable with the lock nuts. If the thread is very clean, it helps.
To adjust the cable, you can space out the cable outers where they engage the trailing arms at the drums by drilling out 1/2" nuts with a 13mm drill, and cutting a slot in the nut to allow it to fit over the cable inner. You can put about 1/2" on each side to space out the outer cable to take up the inner cable slack.
Patience and thin hands are needed. Chapman could not care about after sales maintenance, only the precise engineering. If it was difficult to change, service, or maintain, tough.