I was getting a long boot up time of about 2 minutes.
I ran systemd-analyze blame. (the blame part is cute.)
You should not see apt-daily-service in the listing.
I found a fix that shortened that time by 30 seconds.
This was the tip that worked.
This is Debian bug #844453. apt-daily.service shouldn't be run during boot, but only some time afterward.
As a workaround, do sudo systemctl edit apt-daily.timer and paste the following text into the editor window:
apt-daily timer configuration override
This changes the "timer" that triggers apt-daily.service to run at a random time between 15 min and 45 min after boot, and once a day thereafter. See the systemd.timer manpage for additional (not very well written, alas) explanation of what this means.