While I have not repaired adding machines I have done many mechanical
mechanisms,
For something that complex a manual is very useful for the basic
method of taking assemblies apart.
For some riveted mechanisms immersion in an ultrasonic bath is the
easiest way but any plastics and paint may be damaged.
There are some watch cleaner solutions that dry leaving some oil, they
are designed for the total mechanism immersion in an ultrasonic bath
except for the painted dial.
Else its a careful disassembly and cleaning process, you should be
able to identify sections that can be done in isolation.
Dave Caroline