You step into a big cube. Robot arms tipped with circular saws, knives, and lasers emerge and cut away everything but your central nervous system. Then an emotional-inhibitor implant is attached to the nervous system. Finally, the result is encased in and attached to a suit of cyber-armor, and you can stomp around yelling "Delete!" and "Upgrade!"
It's apparently very painful. There are also robot arms with syringes filled with some kind of green fluid, but this doesn't appear to be anesthetic, because people always scream horribly while it's happening.
There's less information about conversion for the classic series' Mondas Cybermen. It appears that the technology improved gradually over time, instead of being built all at once.
At least sometimes, the Mondas Cybermen expected people to happily embrace conversion, and there wasn't any screaming. So it may be a lot less unpleasant. I'd suggest having them convert you instead of the Cybusmen.