Please specify versions of the Jenkins core and both plugins. Is it a regression after an update?

I added the version numbers.
I don't know whether it is a regression, because I have just started to use both plugins together.

Lars, can you attach a log file from one of the affected builds?

OK, to illustrate it I built a job executing the shell script
for i in $(seq 1 10); do echo time: `date +%H:%M:%S`; sleep 1; done

As you can see the console output should show Timestamper's timestamps next to the timestamps from the shell script. Both timestamps should be equal.
At first the console output with no EnvInject enabled:
20:04:01 Gestartet durch Benutzer
20:04:01 [EnvInject] - Loading node environment variables.
20:04:01 Baue in Arbeitsbereich /var/lib/jenkins/jobs/XYZ/workspace
20:04:01 [workspace] $ /bin/bash -e /tmp/hudson5525335093485642436.sh
20:04:01 time: 20:04:01
20:04:02 time: 20:04:02
20:04:03 time: 20:04:03
20:04:04 time: 20:04:04
20:04:05 time: 20:04:05
20:04:06 time: 20:04:06
20:04:07 time: 20:04:07
20:04:08 time: 20:04:08
20:04:09 time: 20:04:09
20:04:10 time: 20:04:10
20:04:11 Finished: SUCCESS

Everything OK. Then I enabled 'Inject passwords to the build as environment variables' and 'Global passwords' and the output was:
20:03:39 [EnvInject] - Inject global passwords.
20:03:39 [EnvInject] - Mask passwords passed as build parameters.
20:03:39 Gestartet durch Benutzer
20:03:39 [EnvInject] - Loading node environment variables.
20:03:39 Baue in Arbeitsbereich /var/lib/jenkins/jobs/XYZ/workspace
20:03:40 [workspace] $ /bin/bash -e /tmp/hudson2882475315826943329.sh
20:03:41 time: 20:03:39
20:03:42 time: 20:03:40
20:03:43 time: 20:03:41
20:03:44 time: 20:03:42
20:03:45 time: 20:03:43
20:03:46 time: 20:03:44
20:03:47 time: 20:03:45
20:03:48 time: 20:03:46
20:03:49 time: 20:03:47
time: 20:03:48
Finished: SUCCESS

Now the timestamps are shifted.

Hmm, it's just a processing delay. Since you run the job on the Jenkins master, it cannot be caused by communication delays. EnvInject password adds an additional console annotator, so maybe a behavior in the core changes in such case.

Needs to be investigated, but I'm not sure such time difference is a bug. Timestamper annotates logs once it gets info on the master side, so there are data propagation delays by design. Even OS may cause such difference on a high-loaded system

There was no load on the system when I ran the test from my previous comment.

For me these time differences are a bug, because in some cases the timestamps don't help when they are shifted.
And since it's a constant two-line-shift with EnvInject enabled and not a random occurrence, I wouldn't think it's something that's caused by delays or load.