Took me a bit of sleuthing, but this workaround should work, no plugin needed (tested).
1) Grab the iframe “embed” code from the video.
<iframe title="vimeo-player" src="https://player.vimeo.com/video/768688920?h=3e3f80f90c" width="640" height="360" frameborder="0" allowfullscreen></iframe>
2) Add the timestamp to the video URL. Here I’m adding 600s (ie #t=600s):
<iframe title="vimeo-player" src="https://player.vimeo.com/video/768688920?h=3e3f80f90c#t=600s" width="640" height="360" frameborder="0" allowfullscreen></iframe>
3) Now in the block editor (aka Gutenberg), place this in a CUSTOM HTML block. If you use the EMBED block or simply paste the code in the text block, WordPress will automatically convert it to show a preview… but the timestamp won’t work.
Test your video. The timestamp should work now.
But you may realize that the video is not responsive (I got different results with different themes, so do test this).
If you find your video is NOT responsive, remove the CUSTOM HTML block entirely, then:
1) First add a ROW block. You may want to center the row block.
2) Then add a CUSTOM HTML block inside the row block.
3) Finally, put the iframe code inside the CUSTOM HTML block which is inside the ROW block.
Here’s a demo (temp link active for 24 hours only):
https://oon-family-ciye.instawp.link/vimeo-vith-timestamp/
The page has the same video embedded twice, and both of them should have been timestamped to start at 600s (note the blue marker on the players).
The first video was placed in a CUSTOM HTML block in a ROW block and centered. It’s responsive.
The second video was placed in a CUSTOM HTML block directly. It is not responsive.
In case the link goes dead before you read this, I’m adding a screenshot here. Sorry, my phone is in dark mode, but I hope you’ll get the gist of it.
The blue markers on the videos show the timestamped point where the video starts when you hit the play button. And as you can see, the video on top was resized to fit the mobile screen (ie responsive), whereas the video below was not resized.
I hope that helps ??
Good luck!
(And, yes, I get up at 4AM to answer WordPress questions ?? )