You would need two independent inputs at your home in the UK. If you & your wife can watch different programs at home with two (or more) TVs you can input one into slingbox. If not, you can't watch independently via slingbox. In the US most cable or satellite providers provide 2,3 or 4 independent viewing options, perhaps at a small extra cost. With an over-air TV antenna, you would just need a coax cable splitter, possibly with an inexpensive signal booster.
As Ierwin88 mentioned, you need a dual tuner device like a DVR, just confirm that you are able to get different channels at home using two TV's. Basically, the Slingbox will send the IR signal to the cable box but the other TV will be using Ratio Frequency (RF), that's the reason why it will not affect both users