How long the Nationalzation process takes, varies greatly depending on your personal situation.
In general 6 months can be assumed, after which a Einbürgerungszusicherung (assurance of naturalization) will be issued.
In most cases you are then required to relinquish your previous citizenship, which must be completed before the Nationalzation takes place.
Note: Country's that do not allow their citizens to relinquish their citizenship or cause severe disadvantages are known and exceptions are made in these cases.
Otherwise for other countries this process can be very simple, others very time consuming and/or expensive
Assume that you may be Stateless, since some countries revoke it immediately but confirm it only after it goes through their due process.
Should the process take longer than 6 months, assume that the conditions for Nationalzation will be rechecked, which again can take a few months.
After some form of Loss of Nationality Certificate has been issued, a Einbürgerungsurkunde will be handed out.
After the Nationalzation process has ended (you have received a Einbürgerungsurkunde), you must bring the original togeather with a certified translated version of your birth certificate to the Meldebehörde (Registration Office). You should also take some (3-5) photocopies of the Einbürgerungsurkunde with you.
There the change of citizenship will then be ammended to your record. Any photocopies will then be certified.
You may be required to do the same with the Standesamt depending on which state you live in.
After this the original Einbürgerungsurkunde (and the birth certificate) should not be asked for again, since a certified copy should be sufficient.
Otherwise a certified copy of the Einbürgerungsurkunde should be given to any official authorities that know you as a foreigner
- Ausländerbehörde, Job Center (Work Permit) and your present employer
so that they can 'close' their record of you as a foreigner.
For anyone else, the Identity card should be sufficient.
Once the Meldebehörde has changed your record, you can apply for an Identity Card and/or Passport.
Wikipedia says it takes four to six weeks to process, but does it really take that long?
Yes, since they are, just as driver's licenses, printed in the Bundesdruckerei.
See another answer for more details.