Replication Lag on Postgres AWS RDS Service
I had the same issue with our staging RDS. After quite a while I found a blog post that says:


  In a very busy database, with many writes per second, this number will remain fairly accurate. However, in a system where there are few writes, the "replication_delay" will continually grow because the last replayed transaction timestamp isn't increasing.


To test this I wrote a simply script which updates a random record every 0.5s and the lag went to 0.

From the AWS docs under Read Replica Limitations with PostgreSQL - 


  A PostgreSQL Read Replica reports a replication lag of up to five minutes if there are no user transactions occurring on the source DB instance