Question going to be hard to answer w/o knowing resolution you are playing at.
I couldn't find any BBC2 benchies but this will show you how various 5000 series cards perform on AVP and Dirt2 (about 2 and half weeks old)...http://www.hardwareheaven.com/reviews.php?reviewid=937&pageid=7
Testing was performed at 1920x1200 with 2x anti-aliasing and maximum in-game settings.http://www.hardwareheaven.com/reviews.php?reviewid=937&pageid=8
Testing was performed at 1920x1200 with 2x anti-aliasing and ULTRA detail settings.
Based upon those, I'm gonna say a 5830 / 5850 is at least where you wanna be at 1920 x 1200 resolution.....at 1650 x 1050, you should be fine with the 5770.....expect a small performance drop however from your 4870 if using DX10 ..... significant drop in DX11.