There are explicit details on the steps required for this in How do I configure the version of Java that the Apex Language Server uses?
By default, VS Code attempts to locate your local Java installation by looking for a
JDK_HOME environment variable on your computer. If VS Code cannot find your Java installation, or if you want it to use a different installation, change the
Note that while the message specifies the JRE the prerequisites are currently for the JDK.
Upon restarting code.exe wanted network access past the firewall.
The output window for the Apex Language Server also showed messages indicating it was running now.