use Google Cloud Functions with a registry other than registry.npmjs.org

Do Google Cloud Functions not support custom registries?

Google Cloud Functions should support custom registries. What version of Nodejs are you using? There were reported some other similar issues: Runtime doesn't support private npm packages with Nodejs 10.

On the thread it is mentionded that:

I would like to inform you that the Cloud Functions engineering team has rolled out a change to resolve this issue. You may feel free to try your migration efforts from Node.js 8 to Node.js 10 again. The roll out addresses the 404 NOT_FOUND issue when trying to install private packages.

Also I would suggest to test deploying your cloud function using env variables.

gcloud functions deploy functionName 
 --region europe-west1 
 --runtime nodejs10 
 --source=SOURCE
 --set-env-vars= NPM_REGISTRY=<your private registry URL>, NPM_SCOPE=<@private-registry-scope>, NPM_CONFIG_EXTRA=auth_token

EDIT

It seems that the issue resurfaced again for Nodejs10 link