Cloud runs on the exact same underlying compute, storage, and network infrastructure as search, gmail, and other Google services. The big google services are not built on top of cloud themselves (e.g. not inside GCE VMs) partially because they existed before cloud did. Some cloud services are also public versions of existing internal Google services (cloud spanner, cloud bigtable, etc)

Source: work at google, but on search, not cloud.

Are there plans for Google Services (search/gmail/etc) to move GCP anytime soon?

I'm hesitate to use GCP, given that technically Google itself isn't using GCP (totally understanding that a lift and shift of this size takes times, but it would speak volumes if you can tout that Google itself is on GCP)