Also splits the build and publish steps so all packages are built before
first is published, as publishing stomps on package.json files, which
makes subsequent builds trickier.
This needs to be merged to master in order for the action steps to pass.
Build-bot: skip
Test-bot: skip
We don't have permissions to use `npm install --global npm` to get a
specific version, but this action is supported by GH so should be fine.
It remains to be seen if there will be other compatibility issues due to
being a later version of Node than what we are using in v19.
Build-bot: skip
Test-bot: skip
Due to recent changes in NPM package publishing security requirements,
we have to move from TeamCity build to a GitHub Action to publish our
NPM packages, so we can take advantage of trusted publishing. This
change also consolidates and centralizes the npm publishing into
resources/build/ci/npm-publish.sh, which removes a lot of boilerplate
from each of the build.sh scripts, and ensures consistency.
Packages will be `npm pack`ed on PR and test builds, and published in
release builds.
Ref: https://docs.npmjs.com/trusted-publishers
Ref: https://github.blog/changelog/2025-09-29-strengthening-npm-security-important-changes-to-authentication-and-token-management/Fixes: #14963
Test-bot: skip
Build-bot: release:developer
