The short answer is that the E.U. charter, on its face, provides the Right in Title V, Section 1 only with respect to European Union bodies and not from national governments.
It is possible (and indeed, rather likely) that some sort of similar protection may arise via institutions in the government of German or of relevant Lander or municipal governments in Germany. But this right, if it exists, arises either under the German Basic law, or some other statute or ordinance of the relevant German governmental entities, not from the E.U. Charter.
Article 3 of the German Basic Law probably protects your right to be treated impartially and fairly, but not necessarily within a "reasonable time." It could be the timeliness has been interpreted to be encompassed by the other rights in case law, but I have no access to that body of case law.
I suspect, but do not know, that while the law governing the issuance of IDs in Germany is probably federal law, that the administration of that law is probably vested in individual Lander, so the constitution and statutes of the relevant Lander government could be relevant.
It is likely, however, that any right to have your bureaucratic business addressed within a reasonable time in this matter, if it exists at all, is probably a federal or Lander statutory duty, which might even be specific to the precise task you are seeking help with, rather than a general national or Lander constitutional duty.
It is also possible that the right might arise out of some interpretation of one of Germany's treaty obligations in connection with one of the treaties connected to the Council of Europe (which is a distinct institution from the European Union) that guarantees individual rights. My quick scan of the relevant treaties, however, did not identify any parallel rights to the rights one has with respect to E.U. institutions in those treaties.
