Use your central repository where you keep information for all servers (Most likely another SQL server which is used as a central repository for DBA stuff) in the firm. Use below tsql query as a start and modify as necessary. You can loop thru all your servers and collect backup history. That can be another tsql script called via SQL Agent job or PowerShell. Then you will need to set up a threshold for each type of backup and get alerted if the threshold exceeds.
For example if you do a full backup once a week, differential every other days, and transaction log at xx minute interval; you can set up thresholds of 168 hours for full backup, differential 24 hours + some buffer, transaction log xx minute + some buffer.
-- Get Backup History for required database
CAST(CAST(s.backup_size / 1048576 AS INT) AS VARCHAR(14)) AS bkSizeMB,
CAST(CAST(s.compressed_backup_size / 1048576 AS INT) AS VARCHAR(14)) AS Compressed_bkSizeMB,
s.backup_finish_date) AS VARCHAR(12)) + ' ' + 'Seconds' TimeTaken,
CAST(s.first_lsn AS VARCHAR(50)) AS first_lsn,
CAST(s.last_lsn AS VARCHAR(50)) AS last_lsn,
WHEN 'D' THEN 'Full'
WHEN 'I' THEN 'Differential'
WHEN 'L' THEN 'Transaction Log'
END AS BackupType,
FROM msdb.dbo.backupset s
INNER JOIN msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id
WHERE s.backup_start_date>'2017-04-01' --adjust your date
--Uncomment below lines if you want a one or more type of backup
--AND (s.type='D' OR s.type ='I')
--AND (s.type='L' )
--Uncomment below line if you want to filter by database name
--AND database_name ='security'
ORDER BY backup_start_date DESC, backup_finish_date
As an alternative you can leverage these two PowerShell function to achieve the same result.