The 6270A manual shows that the password is 6270. However this does not work in firmware versions 0.39 and lower. For those versions use password 0 (zero) when prompted.
Path to change the date/time is [Setup], <Instrument Setup>, <Instrument Settings>, <Date/Time>, [password prompt].
The password 6270 is implemented in firmware version 0.40 and above.
If that doesn't work, the override password is 9750547