Although there is an accepted answer I don't feel that it is the right one. It is the simplest way to achieve what is needed, but even if already enabled in InnoDB (actually docs tell you that you still should get NULL ...), if you read MySQL docs, even in current version (8.0) using UPDATE_TIME is not the right option, because:
Timestamps are not persisted when the server is restarted or when the
table is evicted from the InnoDB data dictionary cache.
If I understand correctly (can't verify it on a server right now), timestamp gets reset after server restart.
As for real (and, well, costly) solutions, you have Bill Karwin's solution with CURRENT_TIMESTAMP and I'd like to propose a different one, that is based on triggers (I'm using that one).
You start by creating a separate table (or maybe you have some other table that can be used for this purpose) which will work like a storage for global variables (here timestamps). You need to store two fields - table name (or whatever value you'd like to keep here as table id) and timestamp. After you have it, you should initialize it with this table id + starting date (NOW() is a good choice :) ).
Now, you move to tables you want to observe and add triggers AFTER INSERT/UPDATE/DELETE with this or similar procedure:
CREATE PROCEDURE `timestamp_update` ()
BEGIN
UPDATE `SCHEMA_NAME`.`TIMESTAMPS_TABLE_NAME`
SET `timestamp_column`=DATE_FORMAT(NOW(), '%Y-%m-%d %T')
WHERE `table_name_column`='TABLE_NAME';
END