Hi Serge,
Sorry for the slow response. I missed some of these replies. It’s up to 30 days for police background checks and then the digital ID card is sent by diplomatic post so that can take an extra few weeks depending on the timing of the post. 3 months sounds very extreme though. It’s processed by the Estonian Police and Border Guard Board, which is the issuing authority, so people can check on the status of their application at any time by emailing their full name and date of birth to info@politsei.ee.