You may have two buttons widget and two navigation widget ... bind the click event of both buttons to navigate of navigation widget each.
Also as you say web page redirect and not mashups you may need to include iframe on both 2 navigation widget mashups to have source url as your web page
that you need to redirect.
That is an expected behavior. Button widget can't be used for redirecting to a webpage, it is recommended to use hyperlink widget.
Button widget is restricted to trigger a service, however you can write to JS code/function for Button widget, click on JS area (next to Click property) where you can write the function that needs to be triggered. The function script can be written home.js area.
In the othercase, on button click you can execute a TWX service that inturn makes a request to the webpage. I din't actually try this, but may be worth a try.