I had Tsheets for my employees to track hours. They were bought by Intuit, now Quickbooks has bought Intuit. How do I run payroll and import employees hours?

To straighten put your concept, Intuit has owned Quickbooks for decades,  Quickbooks is just one of the platforms and if you looked you might find that the new owner of TSheets is Intuit, the parent company, not Quickbooks.

In any event, from where I sit with both the old and new, only the name and web url has changed. I have not seen any relative change in the layout. 

Each user should have received an email from QuickbooksTime to update from TSheets. Even after they migrate we have had some employees lose access and it has required sending them a new invite from within QuickbooksTime.

The procedures to Approve hours in QT is no different and the sync process should be the same. At least in Desktop the sync button has not moved and there was nothing we has to do within Quickbooks itself to complete the transition 

What you might do, inside the timesheet app,  is click on QuickBooks in the upper right, check your preferences. OR, clear the company path and then reestablish the path