If file system is mounted read only, does relatime option apply? (will it update file access times?)
No. If the filesystem is read-only, new metadata cannot be written to it. This includes access times.
According to the official documentation :
The kernel used in Red Hat Enterprise Linux 7 supports another alternative — relatime. Relatime maintains atime data, but not for each time that a file is accessed. With this option enabled, atime data is written to the disk only if the file has been modified since the atime data was last updated (mtime), or if the file was last accessed more than a certain length of time ago (by default, one day).
We see that the answer is no.