In general, this is called dynamic SQL and it can be done in GBQ with the execute immediate command.

execute immediate (
  select
    string_agg(
      concat(
        'select ',
        chr(39),
        schema_name,
        chr(39),
        ' as schema_name, table_id, date(timestamp_millis(last_modified_time)) as last_modified from project-id.', 
        schema_name,
        '.__TABLES__'
      ),
      ' union all '
    ) as query
  from `project-id`.`region-id`.INFORMATION_SCHEMA.SCHEMATA
)
;

If you have a lot of schemata and are hitting rate limits, you may consider using some kind of loop in a scripting language with delays. Here's a Python/SQL example:

import csv

from google.cloud import bigquery

client = bigquery.Client(project='project-id')
data = []
for dataset in client.list_datasets():
    dataset_id = dataset.dataset_id
    query = f"""
        select
            table_id,
            timestamp_millis(last_modified_time) AS modified
        from `{dataset_id}`.`__TABLES__`
    """
    query_job = client.query(query)
    results = query_job.result()
    for row in results:
        data.append([dataset_id, row.table_id, row.modified])
    # time.sleep(1)

with open('file.csv', 'w', newline='', encoding='utf-8') as csvfile:
    writer = csv.writer(csvfile)
    writer.writerows(data)

Here's a pure Python example:

import csv
import time

from google.cloud import bigquery

client = bigquery.Client(project='project-id')
data = []
for dataset in client.list_datasets():
    for table in client.list_tables(client.dataset(dataset.dataset_id)):
        table = client.get_table(f"{dataset.dataset_id}.{table.table_id}")
        data.append([dataset.dataset_id, table.table_id, table.modified])
        # time.sleep(1)
    # time.sleep(1)

with open('file.csv', 'w', newline='', encoding='utf-8') as csvfile:
    writer = csv.writer(csvfile)
    writer.writerows(data)

You can try this out easily in Google Colab, just add this at the top and make sure that you are signed into an account with access to run jobs and view data.

from google.colab.auth import authenticate_user
authenticate_user()