I'd like @a_horse_with_no_name answer and plus it, but recursive query may be much simpler:
WITH RECURSIVE tree AS (
SELECT i.inhrelid AS oid
FROM pg_inherits i
WHERE i.inhparent = 'scheme.my_table_name'::regclass
UNION ALL
SELECT i.inhrelid AS oid
FROM pg_inherits i
JOIN tree b ON i.inhparent = b.oid
)
SELECT sum(tbl.reltuples)
FROM tree tr
JOIN pg_class tbl ON tr.oid = tbl.oid;
That may be also transformed into a more generic variant for all tables, ready for join (including unpartitioned):
WITH RECURSIVE tables_tree AS ( -- See https://stackoverflow.com/questions/30592826/postgres-approximate-number-of-rows-in-partitioned-tables/68958004#68958004
SELECT oid AS oid, oid as parent_oid
FROM pg_class i
UNION ALL
SELECT i.inhrelid AS oid, t.parent_oid
FROM pg_inherits i
JOIN tables_tree t ON i.inhparent = t.oid
), tables_total_size AS (
SELECT sum(tbl.reltuples) as estimated_table_rows_sum, t.parent_oid
FROM tables_tree t
JOIN pg_class tbl ON t.oid = tbl.oid
GROUP BY t.parent_oid
)
SELECT *, parent_oid::regclass as table_name
FROM tables_total_size
DB-fiddle