I'd like to know what is the most recent versions of jQuery and jQuery-UI that works with Salesforce Lightning with LockerService on?
Any version of jQuery at or above version 2.2.4 should work without any problems. That said, keep in mind that the documentation advises against using DOM manipulation libraries:
In the Lightning approach, the model and the view are decoupled: your code doesn’t have to reach into the DOM. This leads to more robust and more maintainable code. Avoiding direct DOM manipulation is a best practice in the Lightning Component Framework (like in most modern frameworks), and you might just discover that you no longer need your DOM manipulation library.
You may also want to reconsider the use of UI libraries like Bootstrap and jQuery UI. Although these libraries provide useful components, they have their own UI identity that can clash with the Lightning Experience identity. The Base Lightning Components and the Lightning Design System offer similar capabilities while providing a consistent user experience.