Attempts to add a valid npmjs.org version tag to the 'installation' field results in 'Invalid tool ID v0.XX.XX' (respective to the version attempted).Invalid tool ID v0.10.36

Force an update of the update center metadata by clicking 'Update Now' in the Advanced tab of Manage Plugins.

Confirmed resolved by @Daniel Beck's comment

There's an improvement request to update the installer metadata after installing a plugin around somewhere, so yes, if that resolves your specific instance of the problem, nothing else to do here.

Plugin tries to look up a jenkins...json.html file from a mirror.
On that mirror only the corresponding hudson...json.html is available:

java.io.FileNotFoundException: http://mirrors.jenkins-ci.org/updates/updates/jenkins.plugins.nodejs.tools.NodeJSInstaller.json.html?id=jenkins.plugins.nodejs.tools.NodeJSInstaller&version=1.610

Please file a new bug. Since Todd confirmed the workaround solved the problem on his instance, and there's code to ensure compatibility in the plugin (https://github.com/jenkinsci/nodejs-plugin/blob/c4dcc59d8c5be7fb4ed9e7e999b339384f11aa51/src/main/java/jenkins/plugins/nodejs/tools/NodeJSInstaller.java#L385), something else seems to be wrong. Provide much more information, as requested in the wiki:
https://wiki.jenkins-ci.org/display/JENKINS/How+to+report+an+issue

Any suggestion how the 'Update now' could be triggered programmatically, e.g. from a groovy script?

ickersep This is the way I tried and worked.
Get the json file from http://updates.jenkins-ci.org/updates/hudson.plugins.nodejs.tools.NodeJSInstaller.json [1]
Create a file "/var/lib/jenkins/updates/hudson.plugins.nodejs.tools.NodeJSInstaller" [2]
Copy the second parameter json object inside file [1] to the new file [2] created