1.3.7 is identical to version 1.3.6 with the exception that it allows non-premium edition users to access payslips and other reports after the trial period. 1.3.7 was a special edition for existing non-premium users but it is perfectly suitable for premium users too. We'll update the versions page to reflect this.
Your computer clock is absolutely irrelevant from the point of view of PAYE and NI calculations in the software. Otherwise people wouldn't be able to duplicate historic payrolls for example.
What the system looks at is the date of the payroll. From 1.3.6 upwards as you advance a payroll (using the button at the bottom right of the payroll form) from a payroll date before 7/9/08 to a payroll date after 7/9/08 the system will automatically add 60 to L suffix tax codes. If you didn't upgrade your software release before running that particular period-end that took the payroll past the cutoff date then you'll need to increment the tax codes manually.
You can check the tax bands and their dates on the company form.
EDIT: The versions page already showed the information about 1.3.7 but the "What is the current release" question on the My 12Pay page referred to 1.3.6 before directing users to the versions page, so we've altered that text to explain the difference between 1.3.7 and 1.3.6