So, I did some investigation.
If you install OBS (Open Broadcaster Software,
https://objproject.com/download), you can then grab the Visibility Timer script (
https://github.com/cg2121/obs-visibility-timer/releases/download/v0.3/visibility-timer.zip).
Unzip the visibility timer script somewhere, and load up OBS.
It'll ask you a bunch of questions, but you skip most of it.
Click the + in the Sources box, create a new Video Capture Device, selecting your webcam, and then make it as big as possible in the preview screen.
Press the Start Virtual Camera button on the right-hand side. This creates a virtual camera that will display the output from OBS. You can set whatever video app you want to use that virtual camera now instead of your webcam. For example, Chrome can be set to use it by going to Settings (in the menu in the top-right) > Site settings (in the Privacy and security section, you may have to scroll down a bit) > Camera (under permissions) > then select the OBS virtual camera from the drop-down in the top-left. If it's not there, try restarting Chrome and trying again.
Now go to Tools > Scripts, click the +, and select the visibility-timer.lua script you unzipped earlier.
This will show you a bunch of options on the right, select a Mode of 'Show source after specified time', Source should be the Video Capture Device source you created earlier, and then you can set the Duration (in milliseconds) before you want your webcam to be displayed. The timer will start from the moment you type in the Duration you want, but you can reset it by hitting the refresh button (two arrow pointing in a circle) at the bottom left of the Scripts screen.
Now set up whatever application you want to be streaming to - using the OBS virtual camera as the source. When you're ready to start, go back to Tools > Scripts and hit the refresh button - now the timer is running, and you can start streaming, knowing that once the timer has finished, your webcam will become visible.
A bit fiddly, but once you've got most of it set up, it's not too bad. OBS also has a bunch of other things you can do, like add text to the screen to say what is going on, and with some more plugins / scripts, you could even add a visible countdown timer...