In LW click on View > click on Show Download Tray... do your paused downloads show at the bottom of LW now?



If that doesn't help...
You can't resume files with LW 5 like you could with LW 4... any incomplete downloads should show up at the bottom of LW (regardless of whether you have the Home page up or a search page etc) & either resume downloading or give you the option to try again if the download stalls.

I just tried it - had 2 downloads going, paused both, shut down LW & restarted the computer.
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Once I started LW again, the paused downloads automatically connected & resumed until I paused them again.
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In LW go to Tools > Downloads... are any files showing up on that page?