Am Estonian. The remote upgrade system is still working, but only enabled for high-priority users right now because of the high load - medical professionals, social workers, people who used the card more than 100 times over the past 3 months. It will be re-enabled for all users at the start of next week, maybe earlier if the high-priority users have all been serviced.
New certificates are generated on the chip, and the public part is then transmitted to the government public keys directory.