In addition to the repmgr configuration settings, parameters in the conninfo string influence how repmgr makes a network connection to PostgreSQL. In particular, if another server in the replication cluster is unreachable at network level, system network settings will influence the length of time it takes to determine that the connection is not possible.
In particular explicitly setting a parameter for connect_timeout should be considered; the effective minimum value of 2 (seconds) will ensure that a connection failure at network level is reported as soon as possible, otherwise depending on the system settings (e.g. tcp_syn_retries in Linux) a delay of a minute or more is possible.
For further details on conninfo network connection parameters, see the PostgreSQL documentation.