What will happen to the Read replicas, will they continue to serve
requests as the upgrade takes place?
In case of an upgrade a restart is needed for your CloudSQL master instance, your Read replica will remain untouched and it will continue serving the dedicated requests (different than masters).
What will happen to the Read replicas, will they continue to serve
requests as the upgrade takes place? As far as I am aware the read
replicas should have the same specs with the original database. Are
they going to be restarted and upgraded? If not is upgrading the read
replica a manual process?
About Read Replicas specs, master and read replicas must have the same capacity, thus you are only able to change master's capacity which will change also the capacity of the read replicas.
Although this is not the case on the Machine Type (CPU & Memory). You can change CPU and Memory on your read replicas manually as much as you want, but your master instance is able to upgrade memory and CPU until the lower specs of your read replica. So, the master instance is following the smaller read replica.
Take into consideration also Google's documentation for requirements and tips for replication