I use IE at work as a browser but, when on the ATP/WTA/LTA sites and click for live scoring, I get the pop up window but its just a blank black screen.

I would swap to Firefox and it would work fine.  My laptop has just updated and, now the life scoring wont work on Firefox either (just the blank black pop up screen)

Any suggestions how to restore it so I can follow the matches as before?

go into tools, then clear recent history. that sometimes gets the LS working.

that's on firefox, I expect there's an equivalent on IE.

Do you particularly need the separate little box? If not, on the ATP main page, under the tab "scores and stats" there is a "livescore main" heading which brings you to this page: http://www.atpworldtour.com/Scores/Scores-and-Stats.aspx?completed=1&from=topnav

There are tabs for both live and completed matches.

Not sure about women's matches though, there may be something similar on the WTA site.

Or you could try http://www.flashscores.co.uk/tennis/ which gives both.

On Firefox there's the ability to allow pop-ups for certain sites. If they're blocked for all sites then you can unblock for named sites or all sites. So Tools, options, content, that'll enable you to see if the popup blocker applies.