Question: After exporting `HISTTIMEFORMAT='%F %T'` I tried to query `history`

But the result shows all the commands are executed on same day.

How can I check the actual date and time of actual command execution?

Answer: 
If you set the `HISTTIMEFORMAT` in bash your **new** entries get stored in the history file with a timestamp, older commands that don't have a timestamp (those before you ever set `HISTTIMEFORMAT` will display one and the same date-time-stamp (I assume the one from the first entry found with a real timestamp).

This problem should solve itself after your complete history has been updated in a few days.

You can look in `~/.bash_history` to see what is the first line that has a date-time-stamp. Those are lines starting with a `#` followed by a (currently) 10 digit number.
I think this is a feature. As you just changed this setting, you can see from `~/.bash_history` that the old command do not have a time-stamp stored. So for these commands, it will just assume the current time.

Try to put the `export` command in `~/.bashrc` and execute a few command. You will see, that in `~/.bash_history` an additional time-stamp will be save, which can then be displayed by `history`. So for new commands it should work as expected. 

Thus: Not backwards compatible with commands executed in a different terminal window in the past.