I wish people would stop assuming Russell Group universities have some kind of magic power. It's just a self-selected mutual promotional/lobbying group that seeks to divert resources - primarily research funding - to its members and away from other institutions.
Surrey is not a member of the Russell Group. Nor are the Universities of Bath or Leicester, whose research strengths are at least as great as those of one or two Russell Group members. Reading might also be worth a look for biological sciences, although I don't know whether anyone would willingly live in Reading. It depends which particular course you're looking for: different places have different strengths.
Many decades ago, when I was an undergraduate, my college (a constituent of the old federal University of London) was one of very few HE institutions that sent science students to work in industry for 6-month or 12-month placements. Bath & Surrey were two others that did. I heard someone from Surrey give a presentation on the placements programme at a UCAS conference a few years ago; it sounds like it's gone from strength to strength. I've never visited or worked there, though, so I'm only recommending that you look at it.
The high level of unemployment among science graduates is likely to be related to a number of factors, but one is definitely lack of real-world experience in science/technology workplaces. Some students arrange their own experience through summer research projects or short-term paid work in labs. It makes life a bit easier if your university arranges something for you.
LIZS - Southampton is a member of the Russell Group. I used to work there but the building that used to house biological sciences has been knocked down since I left. I'm not sure what happens where now.